A locally finite category is defined as a category in which every arrow admits only finitely many
different ways to be factorized by composable arrows. The large algebra of such categories over
some fields may be defined, and with it a group of invertible series (under multiplication). For
certain particular locally finite categories, a substitution operation, generalizing the usual substitution
of formal power series, may be defined, and with it a group of reversible series (invertible
under substitution). Moreover, both groups are actually affine groups. In this contribution, we
introduce their coordinate Hopf algebras which are both free as commutative algebras. The
semidirect product structure obtained from the action of reversible series on invertible series by
anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras
under the form of a smash coproduct.

Laurent Poinsot

Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions. Moreover, in this way, we are able to compose rooted trees with each other. Such a new composition of rooted trees is a very powerful tool applied in this paper in order to obtain important results as the creation of new rational and Pythagorean trees.

Marco Abrate

Stefano Barbero

Umberto Cerruti

Nadir Murru

Construction and Composition of Rooted Trees via Descent Functions

We present some applications of strict graded categorical groups to the construction of the obstruction of an equivariant kernel and to the classification of equivariant group extensions which are central ones. The composition of a graded categorical group and an equivariant group homomorphism is also determined.

Nguyen Tien Quang

Che Thi Kim Phung

Some Results on Strict Graded Categorical Groups

Let <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M1" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g>

		
	
</svg> a group and <svg     style="vertical-align:-2.3205pt;width:35.037498px;" id="M2" height="15.0875" version="1.1" viewBox="0 0 35.037498 15.0875" width="35.037498"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D714" d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6&#xA;q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" /></g><g transform="matrix(.017,-0,0,-.017,10.908,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,16.789,12.138)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,29.096,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

		
	
</svg> be the set of element orders of <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M3" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg>. Let <svg     style="vertical-align:-2.3205pt;width:62.012501px;" id="M4" height="15.1125" version="1.1" viewBox="0 0 62.012501 15.1125" width="62.012501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g><g transform="matrix(.017,-0,0,-.017,13.322,12.162)"><path id="x2208" d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z" /></g><g transform="matrix(.017,-0,0,-.017,27.023,12.162)"><use xlink:href="#x1D714"/></g><g transform="matrix(.017,-0,0,-.017,37.868,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,43.749,12.162)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,56.056,12.162)"><use xlink:href="#x29"/></g>

		
	
</svg> and let <svg     style="vertical-align:-3.3907pt;width:13.1125px;" id="M5" height="11.9625" version="1.1" viewBox="0 0 13.1125 11.9625" width="13.1125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D460" d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81&#xA;q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" /></g>

			<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><use xlink:href="#x1D458"/></g>

		
	
</svg> be the number of elements of order <svg     style="vertical-align:-0.1638pt;width:8.6750002px;" id="M6" height="12.4375" version="1.1" viewBox="0 0 8.6750002 12.4375" width="8.6750002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D458"/></g>

		
	
</svg> in <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M7" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg>. Let nse<svg     style="vertical-align:-3.3907pt;width:144.2px;" id="M8" height="16.450001" version="1.1" viewBox="0 0 144.2 16.450001" width="144.2"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.162)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,18.251,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,28.841,12.162)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,43.545,12.162)"><path id="x7B" d="M293 -169q-145 0 -145 128q0 35 4 106t3 104q0 17 -2 28t-10 25.5t-28 23t-51 10.5v29q31 2 51 10.5t28 23t10 25.5t2 28q0 31 -4 101t-3 104q0 132 145 132v-28q-51 -7 -67.5 -30t-16.5 -63q0 -32 6.5 -99.5t6.5 -100.5q0 -97 -83 -115v-4q83 -20 83 -117&#xA;q0 -34 -6.5 -99.5t-6.5 -96.5q0 -41 16.5 -65.5t67.5 -31.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,49.444,12.162)"><use xlink:href="#x1D460"/></g>

			<g transform="matrix(.012,-0,0,-.012,55.825,16.25)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,67.163,12.162)"><path id="x2223" d="M162 -163h-61v866h61v-866z" /></g><g transform="matrix(.017,-0,0,-.017,76.342,12.162)"><use xlink:href="#x1D458"/></g><g transform="matrix(.017,-0,0,-.017,89.618,12.162)"><use xlink:href="#x2208"/></g><g transform="matrix(.017,-0,0,-.017,103.319,12.162)"><use xlink:href="#x1D714"/></g><g transform="matrix(.017,-0,0,-.017,114.164,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,120.046,12.162)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,132.353,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,138.234,12.162)"><path id="x7D" d="M283 255q-31 -2 -51 -10.5t-28 -23.5t-10 -27t-2 -30q0 -28 4 -97.5t4 -107.5q0 -128 -146 -128v28q50 7 67.5 31t17.5 63q0 33 -7 100t-7 98q0 96 84 117v4q-84 20 -84 115q0 34 7 102.5t7 98.5q0 40 -17.5 63t-67.5 30v28q28 0 50 -4t45.5 -16.5t37 -40t13.5 -68.5&#xA;q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z" /></g>

		
	
</svg>. In Khatami et al. and Liu's works, <svg     style="vertical-align:-3.3907pt;width:36.087502px;" id="M9" height="16.4125" version="1.1" viewBox="0 0 36.087502 16.4125" width="36.087502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g>

			<g transform="matrix(.012,-0,0,-.012,9.763,16.212)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g>

			<g transform="matrix(.017,-0,0,-.017,16.1,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,21.982,12.138)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g><g transform="matrix(.017,-0,0,-.017,30.141,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> and <svg     style="vertical-align:-3.3907pt;width:36.087502px;" id="M10" height="16.4125" version="1.1" viewBox="0 0 36.087502 16.4125" width="36.087502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43F"/></g>

			<g transform="matrix(.012,-0,0,-.012,9.763,16.212)"><use xlink:href="#x33"/></g>

			<g transform="matrix(.017,-0,0,-.017,16.1,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,21.982,12.138)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g><g transform="matrix(.017,-0,0,-.017,30.141,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> are uniquely determined by nse<svg     style="vertical-align:-2.3205pt;width:24.200001px;" id="M11" height="15.0875" version="1.1" viewBox="0 0 24.200001 15.0875" width="24.200001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,18.251,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg>. In this paper, we prove that if <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M12" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> is a group such that nse<svg     style="vertical-align:-2.3205pt;width:24.200001px;" id="M13" height="15.0875" version="1.1" viewBox="0 0 24.200001 15.0875" width="24.200001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,18.251,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> = nse<svg     style="vertical-align:-3.3907pt;width:51.112499px;" id="M14" height="16.4125" version="1.1" viewBox="0 0 51.112499 16.4125" width="51.112499"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x1D448" d="M751 650l-6 -28q-53 -1 -69 -12.5t-24 -38.5q-23 -85 -43 -192l-13 -69q-34 -181 -130 -262q-79 -65 -185 -65t-171 57q-87 77 -54 258l42 232q12 58 0.5 73t-75.5 19l8 28h263l-6 -28q-63 -5 -78 -18t-26 -74l-41 -226q-25 -135 22 -201t139 -66q99 0 161.5 68t91.5 217&#xA;l11 57q28 149 28 193q0 28 -19 38t-79 12l6 28h247z" /></g>

			<g transform="matrix(.012,-0,0,-.012,18.912,16.212)"><use xlink:href="#x33"/></g>

			<g transform="matrix(.017,-0,0,-.017,25.25,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,31.132,12.138)"><path id="x37" d="M447 623l8 -12l-283 -613l-74 -10l-7 11q174 283 297 551h-216q-48 0 -62.5 -12t-33.5 -63h-29q10 60 18 148h382z" /></g><g transform="matrix(.017,-0,0,-.017,39.291,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,45.173,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg>, then <svg     style="vertical-align:-3.3907pt;width:71.074997px;" id="M15" height="16.4125" version="1.1" viewBox="0 0 71.074997 16.4125" width="71.074997"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,17.078,12.138)"><path id="x2245" d="M535 230h-483v50h483v-50zM535 63h-483v50h483v-50zM536 445q-54 -70 -120 -70q-47 0 -137 43q-78 38 -108 38q-55 0 -91 -57l-31 33q51 74 122 74q45 0 137 -44q76 -37 108 -37q55 0 91 57z" /></g><g transform="matrix(.017,-0,0,-.017,31.782,12.138)"><use xlink:href="#x1D448"/></g>

			<g transform="matrix(.012,-0,0,-.012,44.762,16.212)"><use xlink:href="#x33"/></g>

			<g transform="matrix(.017,-0,0,-.017,51.1,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,56.982,12.138)"><use xlink:href="#x37"/></g><g transform="matrix(.017,-0,0,-.017,65.141,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg>.

Shitian Liu

A Characterization of Projective Special Unitary Group <i>U</i><sub>3</sub>(7) by nse

A module <svg style="vertical-align:-0.0pt;width:17.475px;" id="M3" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g></svg> over an associative ring <svg style="vertical-align:-0.1092pt;width:11.075px;" id="M4" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117&#xA;t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g></svg> with unity is a QTAG module if every finitely generated submodule of any homomorphic image of <svg style="vertical-align:-0.0pt;width:17.475px;" id="M5" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440" /></g></svg> is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M6" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-pure submodules and high submodules are significant. A submodule <svg style="vertical-align:-0.1092pt;width:14.9875px;" id="M7" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g></svg> is quasi-<svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M8" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-dense in <svg style="vertical-align:-0.0pt;width:17.475px;" id="M9" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440" /></g></svg> if <svg style="vertical-align:-2.22495pt;width:37.674999px;" id="M10" height="14.85" version="1.1" viewBox="0 0 37.674999 14.85" width="37.674999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.012)"><use xlink:href="#x1D440" /></g><g transform="matrix(.017,-0,0,-.017,17.418,12.012)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,24.422,12.012)"><path id="x1D43E" d="M764 650l-7 -26q-61 -10 -91.5 -22.5t-76.5 -48.5l-235 -184q104 -152 208 -276q31 -37 54.5 -48.5t68.5 -16.5l-7 -28h-156q-138 170 -213 284q-19 29 -33.5 35t-33.5 -7l-29 -173q-15 -75 -5.5 -90.5t74.5 -20.5l-5 -28h-260l7 28q58 5 74 21.5t30 89.5l70 378&#xA;q12 68 2 83t-72 22l6 28h257l-7 -28q-62 -7 -76 -21.5t-27 -83.5l-32 -172q29 11 78 46q112 84 217 179q18 16 23.5 25.5t-1 14.5t-26.5 8l-30 4l5 28h249z" /></g></svg> is <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M11" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-divisible, for every <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M12" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-pure submodule <svg style="vertical-align:-0.0pt;width:13.3125px;" id="M13" height="11.175" version="1.1" viewBox="0 0 13.3125 11.175" width="13.3125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43E" /></g></svg> of <svg style="vertical-align:-1.59705pt;width:21.35px;" id="M14" height="13.1625" version="1.1" viewBox="0 0 21.35 13.1625" width="21.35" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D440" /></g><g transform="matrix(.017,-0,0,-.017,17.418,11.112)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g></svg> containing <svg style="vertical-align:-0.1638pt;width:18.862499px;" id="M15" height="11.375" version="1.1" viewBox="0 0 18.862499 11.375" width="18.862499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D441" /></g><g transform="matrix(.017,-0,0,-.017,14.919,11.112)"><path id="x2E" d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" /></g></svg> Here we study these submodules and obtain some interesting results. Motivated by <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M16" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-neat envelope, we also define <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M17" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-pure envelope of a submodule <svg style="vertical-align:-0.1092pt;width:14.9875px;" id="M18" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D441" /></g></svg> as the <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M19" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-pure submodule <svg style="vertical-align:-0.546pt;width:47.599998px;" id="M20" height="11.85" version="1.1" viewBox="0 0 47.599998 11.85" width="47.599998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43E" /></g><g transform="matrix(.017,-0,0,-.017,17.962,11.113)"><path id="x2287" d="M531 322q0 -94 -82.5 -160.5t-198.5 -66.5h-193v50h192q93 0 158.5 52t65.5 125q0 74 -65.5 126t-158.5 52h-192v50h193q116 0 198.5 -67t82.5 -161zM502 -40h-445v50h445v-50z" /></g><g transform="matrix(.017,-0,0,-.017,32.666,11.113)"><use xlink:href="#x1D441" /></g></svg> if <svg style="vertical-align:-0.0pt;width:13.3125px;" id="M21" height="11.175" version="1.1" viewBox="0 0 13.3125 11.175" width="13.3125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43E" /></g></svg> has no direct summand containing <svg style="vertical-align:-0.1638pt;width:18.862499px;" id="M22" height="11.375" version="1.1" viewBox="0 0 18.862499 11.375" width="18.862499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D441" /></g><g transform="matrix(.017,-0,0,-.017,14.919,11.112)"><use xlink:href="#x2E" /></g></svg> We find that <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M23" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-pure envelopes of <svg style="vertical-align:-0.1092pt;width:14.9875px;" id="M24" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D441" /></g></svg> have isomorphic basic submodules, and if <svg style="vertical-align:-0.0pt;width:17.475px;" id="M25" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440" /></g></svg> is the direct sum of uniserial modules, then all <svg style="vertical-align:-0.1638pt;width:8.8999996px;" id="M26" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E" /></g></svg>-pure envelopes of <svg style="vertical-align:-0.1092pt;width:14.9875px;" id="M27" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D441" /></g></svg> are isomorphic.

Alveera Mehdi

Fahad Sikander

Firdhousi Begum

On Quasi-<svg style="vertical-align:-0.1638pt" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x210E" d="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24&#xA;q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" /></g></svg>-Dense Submodules and <svg style="vertical-align:-0.1638pt" height="12.4375" version="1.1" viewBox="0 0 8.8999996 12.4375" width="8.8999996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E"/></g></svg>-Pure Envelopes of QTAG Modules

It is well known that the Frattini subgroups of any finite groups are nilpotent. If a finite group is not nilpotent, it is not the Frattini subgroup of a finite group. In this paper, we mainly discuss what kind of finite nilpotent groups cannot be the Frattini subgroup of some finite groups and give some results. Moreover, we generalize Hobby&#x2019;s Theorem.

Qingjun Kong

A Note on Hobby&#x2019;s Theorem of Finite Groups

We generalize the concept of a full term which was introduced by Denecke and Changphas and the concept of the depth of a hypersubstitution which was introduced by  Denecke et al. to the depth of a generalized full term and the depth of a generalized full hypersubstitution and then derive the formular for its depth.

Sarawut Phuapong

Sorasak Leeratanavalee

The Depth of Generalized Full Terms and Generalized Full Hypersubstitutions

We present a new lower bound on the minimum eigenvalue of <svg     style="vertical-align:-0.0pt;width:15.0375px;" id="M2" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43B"/></g>

		
	
</svg>-matrices involving Hadamard products <svg     style="vertical-align:-3.89424pt;width:138.16251px;" id="M3" height="22.387501" version="1.1" viewBox="0 0 138.16251 22.387501" width="138.16251"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,17.475)"><path id="x1D70F" d="M471 456q-22 -60 -37 -85q-20 -3 -84 -3q-9 0 -29 2t-29 2q-38 -145 -54 -234q-15 -80 14 -80q23 0 68 38l14 -25q-78 -83 -137 -83q-50 0 -50 64q0 25 7 57l67 266q-57 4 -98 -9t-80 -49l-20 21q14 18 27 31t36.5 30t56.5 26t73 9q26 0 82.5 -2t89.5 -2q21 0 32 5t24 24&#xA;z" /></g><g transform="matrix(.017,-0,0,-.017,8.273,17.475)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.154,17.475)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" /></g>

			<g transform="matrix(.012,-0,0,-.012,26.612,8.513)"><use xlink:href="#x28"/></g><g transform="matrix(.012,-0,0,-.012,30.729,8.513)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g>

			<g transform="matrix(.008,-0,0,-.008,37.488,11.363)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g>

			<g transform="matrix(.012,-0,0,-.012,42.188,8.513)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g><g transform="matrix(.012,-0,0,-.012,26.612,22.325)"><use xlink:href="#x31"/></g>

			<g transform="matrix(.017,-0,0,-.017,50.712,17.475)"><path id="x2218" d="M326 255q0 -56 -41 -96t-99 -40t-99 40t-41 96t41 96t99 40t99 -40t41 -96zM274 255q0 38 -25.5 65t-62.5 27t-62.5 -27t-25.5 -65t25.5 -65t62.5 -27t62.5 27t25.5 65z" /></g><g transform="matrix(.017,-0,0,-.017,63.615,17.475)"><path id="x22EF" d="M215 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5zM541 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5zM867 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42&#xA;q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5z" /></g><g transform="matrix(.017,-0,0,-.017,86.716,17.475)"><use xlink:href="#x2218"/></g><g transform="matrix(.017,-0,0,-.017,96.796,17.475)"><use xlink:href="#x1D434"/></g>

			<g transform="matrix(.012,-0,0,-.012,109.275,9.313)"><use xlink:href="#x28"/></g><g transform="matrix(.012,-0,0,-.012,113.392,9.313)"><use xlink:href="#x1D6FC"/></g>

			<g transform="matrix(.008,-0,0,-.008,120.15,12.162)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" /></g>

			<g transform="matrix(.012,-0,0,-.012,127.475,9.313)"><use xlink:href="#x29"/></g><g transform="matrix(.012,-0,0,-.012,109.275,21.55)"><use xlink:href="#x1D45A"/></g>

			<g transform="matrix(.017,-0,0,-.017,132.212,17.475)"><use xlink:href="#x29"/></g>

		
	
</svg>, and we show that our lower bound is larger than the lower bound <svg     style="vertical-align:-3.71138pt;width:100.675px;" id="M4" height="18.35" version="1.1" viewBox="0 0 100.675 18.35" width="100.675"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.675)"><path id="x220F" d="M851 711q-62 -5 -76.5 -21.5t-14.5 -78.5v-710q0 -63 14.5 -80t73.5 -21v-28h-277v28q63 4 78 21t15 80v761q0 20 -6 25t-27 5h-358q-23 0 -28.5 -5.5t-5.5 -26.5v-759q0 -63 14 -80t72 -21v-28h-268v28q58 4 72 21t14 80v709q0 63 -14.5 79.5t-75.5 21.5v28h798v-28z&#xA;" /></g>

			<g transform="matrix(.012,-0,0,-.012,18.262,5.388)"><use xlink:href="#x1D45A"/></g><g transform="matrix(.012,-0,0,-.012,18.262,18.138)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g><g transform="matrix(.012,-0,0,-.012,24.248,18.138)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.012,-0,0,-.012,31.232,18.138)"><use xlink:href="#x31"/></g>

			<g transform="matrix(.017,-0,0,-.017,37.562,13.663)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,43.427,13.663)"><use xlink:href="#x1D70F"/></g><g transform="matrix(.017,-0,0,-.017,51.637,13.663)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,57.519,13.663)"><use xlink:href="#x1D434"/></g>

			<g transform="matrix(.012,-0,0,-.012,69.988,17.737)"><use xlink:href="#x1D458"/></g>

			<g transform="matrix(.017,-0,0,-.017,76.587,13.663)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,82.469,13.663)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g>

			<g transform="matrix(.012,-0,0,-.012,88.338,5.5)"><use xlink:href="#x1D6FC"/></g>

			<g transform="matrix(.008,-0,0,-.008,95.1,8.362)"><use xlink:href="#x1D458"/></g>

		
	
</svg>. Three examples verify our result.

Kun Du

Guiding Gu

Guo Liu

Bound on the Minimum Eigenvalue of <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g></svg>-Matrices Involving Hadamard Products

Let <svg     style="vertical-align:-0.0pt;width:15.0375px;" id="M2" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g>

		
	
</svg> be a subgroup of a finite group <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M3" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g>

		
	
</svg>, <svg     style="vertical-align:-3.50804pt;width:10.2px;" id="M4" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g>

		
	
</svg> a prime dividing the order of <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M5" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg>, and <svg     style="vertical-align:-0.0pt;width:10.9375px;" id="M6" height="11.175" version="1.1" viewBox="0 0 10.9375 11.175" width="10.9375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D443" d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5&#xA;t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" /></g>

		
	
</svg> a Sylow <svg     style="vertical-align:-3.50804pt;width:10.2px;" id="M7" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,8.863)"><use xlink:href="#x1D45D"/></g>

		
	
</svg>-subgroup of <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M8" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> for prime <svg     style="vertical-align:-3.50804pt;width:14.075px;" id="M9" height="13.3" version="1.1" viewBox="0 0 14.075 13.3" width="14.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,8.863)"><use xlink:href="#x1D45D"/></g><g transform="matrix(.017,-0,0,-.017,10.143,8.863)"><path id="x2E" d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" /></g>

		
	
</svg> We say that <svg     style="vertical-align:-0.0pt;width:15.0375px;" id="M10" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43B"/></g>

		
	
</svg> is <svg     style="vertical-align:-0.23206pt;width:19.025px;" id="M11" height="11.75" version="1.1" viewBox="0 0 19.025 11.75" width="19.025"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"/></g><g transform="matrix(.017,-0,0,-.017,8.222,11.4)"><use xlink:href="#x1D438"/></g>

		
	
</svg>-supplemented in <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M12" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> if there is a subgroup <svg     style="vertical-align:-0.0pt;width:13.3125px;" id="M13" height="11.175" version="1.1" viewBox="0 0 13.3125 11.175" width="13.3125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43E" d="M764 650l-7 -26q-61 -10 -91.5 -22.5t-76.5 -48.5l-235 -184q104 -152 208 -276q31 -37 54.5 -48.5t68.5 -16.5l-7 -28h-156q-138 170 -213 284q-19 29 -33.5 35t-33.5 -7l-29 -173q-15 -75 -5.5 -90.5t74.5 -20.5l-5 -28h-260l7 28q58 5 74 21.5t30 89.5l70 378&#xA;q12 68 2 83t-72 22l6 28h257l-7 -28q-62 -7 -76 -21.5t-27 -83.5l-32 -172q29 11 78 46q112 84 217 179q18 16 23.5 25.5t-1 14.5t-26.5 8l-30 4l5 28h249z" /></g>

		
	
</svg> of <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M14" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> such that <svg     style="vertical-align:-0.23206pt;width:59.950001px;" id="M15" height="11.75" version="1.1" viewBox="0 0 59.950001 11.75" width="59.950001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,17.078,11.4)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,31.782,11.4)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,46.69,11.4)"><use xlink:href="#x1D43E"/></g>

		
	
</svg> and <svg     style="vertical-align:-4.39398pt;width:101.425px;" id="M16" height="16.65" version="1.1" viewBox="0 0 101.425 16.65" width="101.425"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,18.744,11.112)"><path id="x2229" d="M535 0h-58v354q0 68 -54.5 113.5t-128.5 45.5t-128.5 -45.5t-54.5 -113.5v-354h-58v354q0 90 71.5 149.5t169.5 59.5t169.5 -59.5t71.5 -149.5v-354z" /></g><g transform="matrix(.017,-0,0,-.017,32.496,11.112)"><use xlink:href="#x1D43E"/></g><g transform="matrix(.017,-0,0,-.017,50.413,11.112)"><path id="x2264" d="M531 71l-474 214v50l474 215v-56l-416 -184l416 -183v-56zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,65.1,11.112)"><use xlink:href="#x1D43B"/></g>

			<g transform="matrix(.012,-0,0,-.012,80.025,15.187)"><path id="x1D460" d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81&#xA;q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" /></g><g transform="matrix(.012,-0,0,-.012,84.487,15.187)"><path id="x1D452" d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350&#xA;q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" /></g><g transform="matrix(.012,-0,0,-.012,89.413,15.187)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.012,-0,0,-.012,98.028,15.187)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g>

		
	
</svg> where <svg     style="vertical-align:-3.43848pt;width:33.662498px;" id="M17" height="15.45" version="1.1" viewBox="0 0 33.662498 15.45" width="33.662498"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43B"/></g>

			<g transform="matrix(.012,-0,0,-.012,14.975,15.187)"><use xlink:href="#x1D460"/></g><g transform="matrix(.012,-0,0,-.012,19.437,15.187)"><use xlink:href="#x1D452"/></g><g transform="matrix(.012,-0,0,-.012,24.363,15.187)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> denotes the subgroup of <svg     style="vertical-align:-0.0pt;width:15.0375px;" id="M18" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43B"/></g>

		
	
</svg> generated by all those subgroups of <svg     style="vertical-align:-0.0pt;width:15.0375px;" id="M19" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43B"/></g>

		
	
</svg> which are <svg     style="vertical-align:-0.23206pt;width:8.2875004px;" id="M20" height="11.75" version="1.1" viewBox="0 0 8.2875004 11.75" width="8.2875004"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"/></g>

		
	
</svg>-quasinormally embedded in <svg     style="vertical-align:-0.23206pt;width:16.3125px;" id="M21" height="11.75" version="1.1" viewBox="0 0 16.3125 11.75" width="16.3125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,12.37,11.4)"><use xlink:href="#x2E"/></g>

		
	
</svg> In this paper, we characterize <svg     style="vertical-align:-3.50804pt;width:10.2px;" id="M22" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,8.863)"><use xlink:href="#x1D45D"/></g>

		
	
</svg>-nilpotency and supersolvability of <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M23" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> under the assumption that all maximal subgroups of <svg     style="vertical-align:-0.0pt;width:10.9375px;" id="M24" height="11.175" version="1.1" viewBox="0 0 10.9375 11.175" width="10.9375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D443"/></g>

		
	
</svg> are <svg     style="vertical-align:-0.23206pt;width:19.025px;" id="M25" height="11.75" version="1.1" viewBox="0 0 19.025 11.75" width="19.025"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"/></g><g transform="matrix(.017,-0,0,-.017,8.222,11.4)"><use xlink:href="#x1D438"/></g>

		
	
</svg>-supplemented in <svg     style="vertical-align:-3.43848pt;width:46.787498px;" id="M26" height="16.475" version="1.1" viewBox="0 0 46.787498 16.475" width="46.787498"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g>

			<g transform="matrix(.012,-0,0,-.012,14.925,16.212)"><use xlink:href="#x1D43A"/></g>

			<g transform="matrix(.017,-0,0,-.017,24.162,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,30.044,12.138)"><use xlink:href="#x1D443"/></g><g transform="matrix(.017,-0,0,-.017,40.855,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

		
	
</svg>.

Guo Zhong

Liying Yang

Huaquan Wei

Xuanlong Ma

Jiayi Xia

Finite Groups with Some <svg style="vertical-align:-0.23206pt" height="11.75" version="1.1" viewBox="0 0 19.025 11.75" width="19.025" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g><g transform="matrix(.017,-0,0,-.017,8.222,11.4)"><path id="x1D438" d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5&#xA;l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" /></g></svg>-Supplemented Subgroups

Gelfand-Naimark's theorem states that every commutative <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M3" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space <svg     style="vertical-align:-0.0pt;width:13.5875px;" id="M4" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231&#xA;l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z" /></g>

		
	
</svg>, <svg     style="vertical-align:-3.56265pt;width:23.4625px;" id="M5" height="16.6625" version="1.1" viewBox="0 0 23.4625 16.6625" width="23.4625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D6FD" d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5&#xA;t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" /></g><g transform="matrix(.017,-0,0,-.017,9.939,12.162)"><use xlink:href="#x1D44B"/></g>

		
	
</svg> is dense-<svg     style="vertical-align:-0.1638pt;width:9.7749996px;" id="M6" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g>

		
	
</svg>-separable if and only if <svg     style="vertical-align:-2.3205pt;width:37.337502px;" id="M7" height="15.0875" version="1.1" viewBox="0 0 37.337502 15.0875" width="37.337502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,12.047,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,17.928,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,31.391,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

		
	
</svg> is <svg     style="vertical-align:-0.1638pt;width:9.7749996px;" id="M8" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D6FC"/></g>

		
	
</svg>-cogenerated if and only if every family of maximal ideals of <svg     style="vertical-align:-2.3205pt;width:37.337502px;" id="M9" height="15.0875" version="1.1" viewBox="0 0 37.337502 15.0875" width="37.337502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,12.047,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,17.928,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,31.391,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> with zero intersection has a subfamily with cardinal number less than <svg     style="vertical-align:-0.1638pt;width:9.7749996px;" id="M10" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D6FC"/></g>

		
	
</svg> and zero intersection. This gives a simple characterization of <svg     style="vertical-align:-0.1638pt;width:9.7749996px;" id="M11" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D6FC"/></g>

		
	
</svg>-cogenerated commutative unital <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M12" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-algebras via their maximal ideals.

Ehsan Momtahan

On <svg style="vertical-align:-0.1638pt" height="8.1625004" version="1.1" viewBox="0 0 10.7375 8.1625004" width="10.7375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.9)"><path id="x1D736" d="M505 450h125l-163 -259q-1 -4 -1 -12q0 -19 1 -23q9 -98 41 -98q36 0 58 76l24 -7q-34 -139 -121 -139q-32 0 -57.5 24t-31.5 55q-72 -77 -181 -77q-73 0 -114 44t-41 124q0 118 79 210q79 93 197 93q51 0 86 -27t50 -71zM369 281q0 141 -62 141q-43 0 -95 -84&#xA;q-52 -82 -52 -195q0 -50 15 -83t42 -33q35 0 79 57q43 58 67 121q6 35 6 76z" /></g></svg>-Cogenerated Commutative Unital <svg style="vertical-align:-0.23206pt" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.688)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5&#xA;q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g></svg>-Algebras

The famous Poincar&#xe9;-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds
into its associative envelope&#x2014;its universal enveloping algebra&#x2014;as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way&#x2014;less known&#x2014;to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation <svg     style="vertical-align:-0.1638pt;width:48.262501px;" id="M1" height="19.0375" version="1.1" viewBox="0 0 48.262501 19.0375" width="48.262501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,18.213)"><path id="x1D44E" d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387&#xA;q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" /></g><g transform="matrix(.017,-0,0,-.017,13.935,18.213)"><path id="x21A6" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-677v-154h-58v358h58v-154h677l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.017,-0,0,-.017,34.877,18.213)"><use xlink:href="#x1D44E"/></g>

			<g transform="matrix(.012,-0,0,-.012,43.488,10.05)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" /></g>

		
	
</svg>, that is, a commutative differential algebra, admits a Wronskian bracket <svg     style="vertical-align:-2.3205pt;width:131.97501px;" id="M2" height="21.7125" version="1.1" viewBox="0 0 131.97501 21.7125" width="131.97501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.625,18.213)"><path id="x1D44A" d="M1004 650l-6 -29q-54 -6 -71 -19.5t-51 -74.5l-271 -539h-33l-98 506h-3l-258 -506h-30l-78 532q-10 67 -21.5 80t-66.5 21l6 29h241l-8 -29l-26 -5q-34 -6 -41 -16t-3 -47l59 -425h4l251 510h31l102 -510h2q150 299 198 423q14 40 8.5 48.5t-47.5 16.5l-28 5l7 29h231z&#xA;" /></g><g transform="matrix(.017,-0,0,-.017,17.896,18.213)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,23.777,18.213)"><use xlink:href="#x1D44E"/></g><g transform="matrix(.017,-0,0,-.017,32.379,18.213)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,39.076,18.213)"><path id="x1D44F" d="M452 333q0 -82 -46 -164t-116 -128q-81 -53 -158 -53q-26 0 -51 13t-37 32q-26 41 -6 134l91 431q7 40 2.5 47t-34.5 7h-33l2 25q36 4 72.5 13t58.5 15.5t28 6.5q10 0 4 -29l-91 -391h2q65 77 130 116.5t105 39.5q36 0 56.5 -32t20.5 -83zM365 316q0 76 -35 76&#xA;q-31 0 -93.5 -49t-107.5 -110q-12 -37 -17 -70q-9 -65 12 -96.5t59 -31.5q31 0 56 14q53 29 89.5 105t36.5 162z" /></g><g transform="matrix(.017,-0,0,-.017,47.049,18.213)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g><g transform="matrix(.017,-0,0,-.017,57.656,18.213)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,72.36,18.213)"><use xlink:href="#x1D44E"/></g><g transform="matrix(.017,-0,0,-.017,80.961,18.213)"><use xlink:href="#x1D44F"/></g>

			<g transform="matrix(.012,-0,0,-.012,88.938,10.05)"><use xlink:href="#x2032"/></g>

			<g transform="matrix(.017,-0,0,-.017,96.875,18.213)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,110.61,18.213)"><use xlink:href="#x1D44E"/></g>

			<g transform="matrix(.012,-0,0,-.012,119.225,10.05)"><use xlink:href="#x2032"/></g>

			<g transform="matrix(.017,-0,0,-.017,123.388,18.213)"><use xlink:href="#x1D44F"/></g>

		
	
</svg> under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

Wronskian Envelope of a Lie Algebra

This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get &#x201c;almost&#x201d; Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions.

Gabriel Picavet

Recent Progress on Submersions: A Survey and New Properties

The cyclic graph <svg     style="vertical-align:-3.43848pt;width:17.975px;" id="M1" height="15.45" version="1.1" viewBox="0 0 17.975 15.45" width="17.975"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x393" d="M485 501l-30 -4q-15 64 -35 90q-12 16 -33.5 22t-68.5 6h-74q-21 0 -26.5 -5t-5.5 -26v-458q0 -62 15.5 -78t83.5 -20v-28h-270v28q58 4 72 20t14 78v399q0 61 -13.5 77t-71.5 20v28h433q0 -51 10 -149z" /></g>

			<g transform="matrix(.012,-0,0,-.012,8.675,15.187)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g>

		
	
</svg> of a finite group <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M2" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> is
as follows: take <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M3" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> as the vertices of <svg     style="vertical-align:-3.43848pt;width:17.975px;" id="M4" height="15.45" version="1.1" viewBox="0 0 17.975 15.45" width="17.975"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x393"/></g>

			<g transform="matrix(.012,-0,0,-.012,8.675,15.187)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> and join two distinct
vertices <svg     style="vertical-align:-0.1638pt;width:9.625px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 9.625 7.9499998" width="9.625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g>

		
	
</svg> and <svg     style="vertical-align:-3.56265pt;width:9.9624996px;" id="M6" height="12.175" version="1.1" viewBox="0 0 9.9624996 12.175" width="9.9624996"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D466" d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26&#xA;q13 28 43 28q18 0 30 -15.5t12 -39.5z" /></g>

		
	
</svg> if <svg     style="vertical-align:-3.56265pt;width:40.224998px;" id="M7" height="16.512501" version="1.1" viewBox="0 0 40.224998 16.512501" width="40.224998"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x27E8" d="M360 -163h-62l-192 430v6l192 430h62l-193 -433z" /></g><g transform="matrix(.017,-0,0,-.017,7.083,12.012)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,16.585,12.012)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,23.283,12.012)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,33.125,12.012)"><path id="x27E9" d="M307 267l-192 -430h-62l193 433l-193 433h62l192 -430v-6z" /></g>

		
	
</svg> is cyclic. In this paper, we
investigate how the graph theoretical properties of <svg     style="vertical-align:-3.43848pt;width:17.975px;" id="M8" height="15.45" version="1.1" viewBox="0 0 17.975 15.45" width="17.975"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x393"/></g>

			<g transform="matrix(.012,-0,0,-.012,8.675,15.187)"><use xlink:href="#x1D43A"/></g>

		
	
</svg> affect the group theoretical
properties of <svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M9" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D43A"/></g>

		
	
</svg>. First, we consider some properties of <svg     style="vertical-align:-3.43848pt;width:17.975px;" id="M10" height="15.45" version="1.1" viewBox="0 0 17.975 15.45" width="17.975"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x393"/></g>

			<g transform="matrix(.012,-0,0,-.012,8.675,15.187)"><use xlink:href="#x1D43A"/></g>

		
	
</svg>
and characterize certain finite groups whose cyclic graphs have some properties. 
Then, we present some properties of the cyclic graphs of
the dihedral groups <svg     style="vertical-align:-3.3907pt;width:26.0875px;" id="M11" height="15.4" version="1.1" viewBox="0 0 26.0875 15.4" width="26.0875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D437" d="M780 372q0 -191 -178 -297q-128 -75 -335 -75h-250l7 27q61 7 77 22t28 76l77 398q11 62 0 76.5t-77 22.5l6 28h292q164 0 258.5 -67.5t94.5 -210.5zM678 375q0 119 -73 180.5t-201 61.5q-62 0 -82 -13q-17 -7 -25 -53l-80 -427q-8 -47 -4 -62t26 -20q24 -6 80 -6&#xA;q166 0 262.5 95t96.5 244z" /></g>

			<g transform="matrix(.012,-0,0,-.012,13.712,15.187)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g><g transform="matrix(.012,-0,0,-.012,19.424,15.187)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g>

		
	
</svg> and the generalized quaternion groups <svg     style="vertical-align:-3.3907pt;width:25.487499px;" id="M12" height="15.6875" version="1.1" viewBox="0 0 25.487499 15.6875" width="25.487499"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D444" d="M745 361q0 -134 -83.5 -233t-214.5 -130l16 -11q97 -67 250 -132l-8 -23q-76 3 -131 16q-81 19 -242 125l-20 13q-129 8 -209 91t-80 208q0 160 116 271t289 111q136 0 226.5 -83t90.5 -223zM645 356q0 127 -57.5 201.5t-169.5 74.5q-126 0 -210.5 -104.5t-84.5 -248.5&#xA;q0 -97 46 -166.5t129 -87.5l84 15l29 -19q104 21 169 121.5t65 213.5z" /></g>

			<g transform="matrix(.012,-0,0,-.012,13.113,15.475)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g><g transform="matrix(.012,-0,0,-.012,18.824,15.475)"><use xlink:href="#x1D45B"/></g>

		
	
</svg> for some <svg     style="vertical-align:-0.1638pt;width:8.6625004px;" id="M13" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"/></g>

		
	
</svg>. 
Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of
order up to <svg     style="vertical-align:-0.0pt;width:16.4375px;" id="M14" height="10.9125" version="1.1" viewBox="0 0 16.4375 10.9125" width="16.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,8.222,10.862)"><use xlink:href="#x34"/></g>

		
	
</svg>.

Xuan Long Ma

Hua Quan Wei

The Cyclic Graph of a Finite Group

The notion of intersectional soft subalgebras of a BE-algebra is introduced, and related properties are investigated. Characterization of an intersectional soft subalgebra is discussed. The problem of classifying intersectional soft subalgebras by their inclusive subalgebras will be solved.

Young Bae Jun

Sun Shin Ahn

Applications of Soft Sets in <i>BE</i>-Algebras

We provide a recipe for &#x201c;fattening&#x201d; a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category.

Saikat Chatterjee

Amitabha Lahiri

Ambar N. Sengupta

A Morphism Double Category and Monoidal Structure

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Algebra