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ISRN Algebra
VolumeΒ 2011Β (2011), Article IDΒ 851495, 8 pages
http://dx.doi.org/10.5402/2011/851495
Research Article

Finite Groups Whose Certain Subgroups of Prime Power Order Are 𝑆-Semipermutable

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 8o111, Jeddah 21589, Saudi Arabia

Received 17 July 2011; Accepted 3 August 2011

Academic Editor: A. Kiliçman

Copyright Β© 2011 Mustafa Obaid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is said to be S-semipermutable in 𝐺 if 𝐻 permutes with every Sylow 𝑝-subgroup of 𝐺 with (𝑝,|𝐻|)=1. In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

1. Introduction

All groups considered in this paper will be finite. Two subgroups 𝐻 and 𝐾 of a group 𝐺 are said to permute if 𝐻𝐾=𝐾𝐻. It is easily seen that 𝐻 and 𝐾 permute if and only if 𝐻𝐾 is a subgroup of 𝐺. We say, following Kegel [1], that a subgroup of 𝐺 is 𝑆-quasinormal in 𝐺 if it permutes with every Sylow subgroup of 𝐺. Chen [2] introduced the following concept: a subgroup 𝐻 of 𝐺 is said to be 𝑆-semipermutable in 𝐺 if 𝐻 permutes with every Sylow 𝑝-subgroup of 𝐺 with (𝑝,|𝐻|)=1. Obviously, every 𝑆-quasinormal subgroup of 𝐺 is an 𝑆-semipermutable subgroup of 𝐺. In contrast to the fact that every 𝑆-quasinormal sub-group of 𝐺 is a subnormal subgroup of 𝐺 (see [1]), it does not hold in general that every 𝑆-semipermutable subgroup of 𝐺 is a subnormal subgroup of 𝐺. It suffices to consider the alternating group of degree 4.

Several authors have investigated the structure of a finite group when some information is known about some subgroups of prime power order in the group. Huppert [3] proved that a finite group 𝐺 is solvable provided that all subgroups of prime order are normal in 𝐺. Buckley [4], proved that a group 𝐺 of odd order is supersolvable provided that all subgroups of prime order are normal in 𝐺. Srinivasan [5], and proved that a finite group 𝐺 is supersolvable if the maximal subgroups of every Sylow subgroup of 𝐺 are normal in 𝐺. Developing the result of Srinivasan, Ramadan [6] proved that if 𝐺 is a solvable group and the maximal subgroups of every Sylow subgroup of the Fitting subgroup 𝐹(𝐺) of 𝐺 are normal in 𝐺, then 𝐺 is supersolvable.

For a finite 𝑝-group 𝑃, we denote Ξ©(𝑃)=Ξ©1(𝑃)if𝑝>2,Ξ©(𝑃)=⟨Ω1(𝑃),Ξ©2(𝑃)⟩if𝑝=2,(1.1) where Ω𝑖(𝑃)=⟨π‘₯βˆˆπ‘ƒβˆΆ|π‘₯|=π‘π‘–βŸ©.

Of late there has been a considerable interest to investigate the influence of the abelian subgroups of largest possible exponent of prime power order (we call such subgroups ALPE-subgroups) on the structure of the group. Asaad et al. [7] proved that if 𝐺 is a group such that for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺𝑝 of 𝐺, the ALPE-subgroups of 𝐺𝑝 (resp., Ξ©(𝐺𝑝)) are normal in 𝐺, then 𝐺 is supersolvable. Ramadan [8] proved the following two results. (1) Let 𝐺 be a group such that for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺𝑝 of 𝐺, the ALPE-subgroups of 𝐺𝑝 (resp., Ξ©(𝐺𝑝)) are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable. (2) Let 𝐾 be a normal subgroup of 𝐺 such that 𝐺/𝐾 is supersolvable. If for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐾𝑝 of 𝐾, the ALPE-subgroups of 𝐾𝑝 (resp., Ξ©(𝐾𝑝)) are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable.

In this paper, we study the structure of a finite group under the assumption that certain subgroups of prime power order are 𝑆-semipermutable in the group. We focus our attention on 𝑆-semipermutability property of the ALPE-subgroups of a fixed ALPE-subgroup having maximal order of the Sylow subgroups of a finite group. Furthermore, we improve and extend the above-mentioned results by using the concept of 𝑆-semipermutability.

2. Preliminaries

In this section, we give some results which will be useful in the sequal.

Lemma 2.1 (see [2, Lemmas 1 and 2]). Let 𝐺 be a group.(i)If 𝐻 is a 𝑆-semipermutable subgroup of 𝐺 and 𝐾 is a subgroup of 𝐺 such that 𝐻≀𝐾≀𝐺, then 𝐻 is 𝑆-semipermutable in 𝐾.(ii)Let πœ‹ be a set of primes, 𝑁 a normal πœ‹ξ…ž-subgroup of 𝐺, and 𝐻 a πœ‹-subgroup of 𝐺. If 𝐻 is 𝑆-semipermutable in 𝐺, then 𝐻𝑁/𝑁 is 𝑆-semipermutable in 𝐺/𝑁.

Lemma 2.2 (see [9, Lemma A]). Let 𝐻 be a 𝑝-subgroup of 𝐺; for some prime 𝑝. Then 𝐻 is 𝑆-quasinormal in 𝐺 if and only if 𝑂𝑝(𝐺)≀𝑁𝐺(𝐻), where 𝑂𝑝(𝐺) is the normal subgroup of 𝐺 generated by all π‘ξ…ž-elements of 𝐺.

Lemma 2.3. Let 𝐻 be a 𝑝-subgroup of 𝐺, 𝑝 is a prime. Then the following statements are equivalent:(i)𝐻 is 𝑆-quasinormal in 𝐺;(ii)𝐻≀𝑂𝑝(𝐺) and 𝐻 is 𝑆-semipermutable in 𝐺.

Proof. (i)β‡’(ii): Suppose that 𝐻 is 𝑆-quasinormal in 𝐺. So it follows by [1, Satz 1, page 209] that 𝐻 is subnormal in 𝐺 and then by [10, Lemma 8.6(a), page 28] that 𝐻≀𝑂𝑝(𝐺). Since 𝐻 is 𝑆-quasinormal in 𝐺, obviously, it is 𝑆-semipermutable in 𝐺. Thus (ii) holds.
(ii)β‡’(i): Since 𝐻 is 𝑆-semipermutable in 𝐺, then π»πΊπ‘ž=πΊπ‘žπ» for every Sylow π‘ž-subgroup πΊπ‘ž of 𝐺 with (π‘ž,|𝐻|)=1. Clearly, 𝐻=𝑂𝑝(𝐺)βˆ©π»πΊπ‘ž is normal in π»πΊπ‘ž and so πΊπ‘žβ‰€π‘πΊ(𝐻). Thus 𝑂𝑝(𝐺)≀𝑁𝐺(𝐻). Applying Lemma 2.2, we have that 𝐻 is 𝑆-quasinormal in 𝐺. Thus (i) holdes.

Lemma 2.4 (see [7, Theorem 4, page 253]). Let 𝑃 be a normal 𝑝-subgroup of 𝐺. If the ALPE-subgroups of 𝑃 are normal in 𝐺, then 𝑃 is supersolvably embedded in 𝐺.

Lemma 2.5 (see [11, Lemma 3.8, page 2245]). Let 𝑝 be the smallest prime dividing the order of a group 𝐺, and let 𝐺𝑝 be a Sylow 𝑝-subgroup of 𝐺. If Ξ©(𝐺𝑝)≀genz∞(𝐺), then 𝐺 is 𝑝-nilpotent.

Lemma 2.6 (see [12, Lemma 2.6]). Let 𝑁 be a nontrivial normal subgroup of a group 𝐺. If π‘βˆ©Ξ¦(𝐺)=1, then the Fitting subgroup 𝐹(𝑁) of 𝑁 is the direct product of minimal normal subgroups of 𝐺 which are contained in 𝐹(𝑁).

Lemma 2.7 (see [13, Lemma 3.3.1, page 23]). Suppose that 𝐺𝑝 is a normal Sylow 𝑝-subgroup of 𝐺 and that Ξ©(𝐺𝑝)𝐾 is supersolvable, where 𝐾 is a π‘ξ…ž-Hall subgroup of 𝐺. Then 𝐺 is supersolvable.

3. Main Results

Theorem 3.1. Let 𝑝 be the smallest prime dividing the order of a group 𝐺, and let 𝐺𝑝 be a Sylow 𝑝-subgroup of 𝐺. Fix an ALPE-subgroup 𝑃 of 𝐺𝑝 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is 𝑝-nilpotent.

Proof. Suppose that the theorem is false, and let 𝐺 be a counterexample of minimal order. We prove the following steps.
(1) If 𝑃≀𝑀<𝐺, Then 𝑀 Is 𝑝-Nilpotent
It is clear to see by Lemma 2.1 that the ALPE-subgroups of 𝑃 are 𝑆-semipermutale in 𝑀, so that 𝑀 satisfies the hypothesis of the theorem. Thus, the minimality of 𝐺 yields that 𝑀 is 𝑝-nilpotent.

(2) 𝑁𝐺(𝑃) Is 𝑝-Nilpotent
Suppose that 𝑃 is normal in 𝐺. Let 𝐻 be an ALPE-subgroup of 𝑃 (in particular, we may take 𝐻=𝑃). By hypothesis, 𝐻 is 𝑆-semipermutable in 𝐺 and so by Lemma 2.3, we have that 𝐻 is 𝑆-quasinormal in 𝐺. Hence π»πΊπ‘ž is a subgroup of 𝐺, where πΊπ‘ž is a Sylow π‘ž-subgroup of 𝐺 with π‘žβ‰ π‘. Clearly, 𝐻 is a subnormal Hall subgroup of π»πΊπ‘ž. Thus 𝐻 is normal in π»πΊπ‘ž and hence 𝐻 is normal in π‘ƒπΊπ‘ž as 𝑃 is abelian. Thus 𝑃 is supersolvably embedded in π‘ƒπΊπ‘ž by Lemma 2.4 and so π‘ƒβ‰€π‘„βˆž(π‘ƒπΊπ‘ž). Since π‘„βˆž(π‘ƒπΊπ‘ž)≀genz∞(π‘ƒπΊπ‘ž) by [14, page 34], it follows by Lemma 2.5 that π‘ƒπΊπ‘ž is 𝑝-nilpotent. Thus π‘ƒπΊπ‘ž=π‘ƒΓ—πΊπ‘ž. Hence πΊπ‘žβ‰€πΆπΊ(𝑃), so that 𝑂𝑝(𝐺)≀𝐢𝐺(𝑃).If𝐢𝐺(𝑃)<𝐺, then 𝐢𝐺(𝑃) is 𝑝-nilpotent by (1). Thus 𝑂𝑝(𝐺) is 𝑝-nilpotent and so 𝐺 is 𝑝-nilpotent: a contradiction. Thus we may assume that 𝐢𝐺(𝑃)=𝐺. Then 𝑃≀𝑍(𝐺), in particular, 𝑃≀𝑍(𝐺𝑝). So, 𝑃=𝐺𝑝 by the maximality of 𝑃 and we have by [15, Theorem 4.3, page 252] that 𝐺 is 𝑝-nilpotent: a contradiction. Thus we may assume that 𝑁𝐺(𝑃)<𝐺. According to (1), we have that 𝑁𝐺(𝑃) is 𝑝-nilpotent.

(3) 𝑂𝑝′(𝐺)=1
If 𝑂𝑝′(𝐺)β‰ 1, we consider the quotient group 𝐺/𝑂𝑝′(𝐺). Clearly, 𝐺𝑝𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) is a Sylow 𝑝-subgroup of 𝐺/𝑂𝑝′(𝐺) and 𝑃𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) is an ALPE-Subgroup of 𝐺𝑝𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) having maximal order. By Lemma 2.1, the hypotheses are inherited over 𝐺/𝑂𝑝′(𝐺). Thus, the minimality of 𝐺 implies that 𝐺/𝑂𝑝′(𝐺) is 𝑝-nilpotent, hence 𝐺 is 𝑝-nilpotent, which is a contradiction.

(4) 𝐺=πΊπ‘πΊπ‘ž, Where πΊπ‘ž Is a Sylow π‘ž-Subgroup of 𝐺 with π‘žβ‰ π‘
Since 𝐺 is not 𝑝-nilpotent by [15, Theorem 4.5, page 253], there exists a subgroup 𝐻 of 𝐺𝑝 such that 𝑁𝐺(𝐻) is not 𝑝-nilpotent. But 𝑁𝐺(𝐺𝑝) is 𝑝-nilpotent by a similar argument of the proof of the step (2). Thus we may choose a subgroup 𝐻 of 𝐺𝑝 such that 𝑁𝐺(𝐻) is not 𝑝-nilpotent but 𝑁𝐺(𝐾) is 𝑝-nilpotent for every subgroup 𝐾 of 𝐺𝑝 with 𝐻<𝐾≀𝐺𝑝. It is easy to see that 𝑁𝐺(𝐺𝑝)≀𝑁𝐺(𝐻)≀𝐺. If 𝑁𝐺(𝐻)<𝐺, it follows by (1) that 𝑁𝐺(𝐻) is 𝑝-nilpotent: a contradiction. Thus 𝑁𝐺(𝐻)=𝐺. This leads to 𝑂𝑝(𝐺)β‰ 1 and 𝑁𝐺(𝐾) is 𝑝-nilpotent for every subgroup 𝐾 of 𝐺𝑝 with 𝑂𝑝(𝐺)<𝐾≀𝐺𝑝. Now, by [15, Theorem 4.5, page 253] again, we see that 𝐺/𝑂𝑝(𝐺) is 𝑝-nilpotent and therefore that 𝐺 is 𝑝-solvable. Since 𝐺 is 𝑝-solvable, for any π‘žβˆˆπœ‹(𝐺) with π‘žβ‰ π‘, there exists a Sylow π‘ž-subgroup πΊπ‘ž of 𝐺 such that πΊπ‘πΊπ‘žβ‰€πΊ by [15, Theorem 3.5, page 229]. If πΊπ‘πΊπ‘ž<𝐺, then πΊπ‘πΊπ‘ž is 𝑝-nilpotent by (1) and hence 𝑂𝑝(𝐺)πΊπ‘ž is 𝑝-nilpotent. Thus 𝑂𝑝(𝐺)πΊπ‘ž=𝑂𝑝(𝐺)Γ—πΊπ‘ž. This leads to πΊπ‘žβ‰€πΆπΊ(𝑂𝑝(𝐺))≀𝑂𝑝(𝐺) by [15, Theorem 3.2, page 228] as 𝑂𝑝′(𝐺)=1 by (3), which is a contradiction. Thus 𝐺=πΊπ‘πΊπ‘ž.

(5) The Final Contradiction
Let 𝑁 be a minimal normal subgroup of 𝐺 such that 𝑁≀𝑂𝑝(𝐺). Clearly, π‘βˆ©π‘(𝐺𝑝)β‰ 1 and so 𝑍(𝐺𝑝)≀𝑃 by the maximality of 𝑃. Hence 1β‰ π‘βˆ©π‘(𝐺𝑝)β‰€π‘βˆ©π‘ƒ. By hypothesis, π‘ƒπΊπ‘žβ‰€πΊ for any Sylow π‘ž-subgroup πΊπ‘ž of 𝐺 with (π‘ž,|𝑃|)=1. It is easy to see that π‘βˆ©π‘ƒ=π‘βˆ©π‘ƒπΊπ‘žβŠ²π‘ƒπΊπ‘ž. Thus 𝑂𝑝(𝐺)≀𝑁𝐺(π‘βˆ©π‘ƒ). If 𝑁𝐺(π‘βˆ©π‘ƒ)<𝐺, then by (1), 𝑁𝐺(π‘βˆ©π‘ƒ) is 𝑝-nilpotent. Hence 𝑂𝑝(𝐺) is 𝑝-nilpotent and so also does 𝐺: a contradiction. Thus we may assume that 𝑁𝐺(π‘βˆ©π‘ƒ)=𝐺. By the minimality of 𝑁 and since π‘βˆ©π‘ƒβ‰ 1, we have that π‘βˆ©π‘ƒ=𝑁 and so 𝑁≀𝑃. If π‘ƒπΊπ‘ž<𝐺, then π‘ƒπΊπ‘ž is 𝑝-nilpotent by (1) and hence π‘πΊπ‘ž is 𝑝-nilpotent. Thus π‘πΊπ‘ž=π‘Γ—πΊπ‘ž and so πΊπ‘žβ‰€πΆπΊ(𝑁). Thus by (4), 𝐺/𝐢𝐺(𝑁) is a 𝑝-group and so by [14, Theorem 6.3, page 221], π‘β‰€π‘βˆž(𝐺). Since π‘βˆž(𝐺)β‰€π‘„βˆž(𝐺), we have that π‘β‰€π‘„βˆž(𝐺) which implies that 𝑁 is supersolvably embedded in 𝐺 and so clearly that |𝑁|=𝑝. Thus, it is easy to see that the quotient group 𝐺/𝑁 satisfies the hypothesis of the theorem by Lemma 2.1. Now, by the minimality of 𝐺, we see that 𝐺/𝑁 is 𝑝-nilpotent. Since the class of all 𝑝-nilpotent groups is a saturated formation, we have that 𝑁 is the unique minimal normal subgroup of 𝐺 and 𝑁/β©½Ξ¦(𝐺). Thus Ξ¦(𝐺)=1 and hence 𝑁=𝑂𝑝(𝐺) by Lemma 2.6 and so 𝐹(𝐺)=𝑂𝑝(𝐺)=𝑁 by (3). Hence πΊπ‘žβ‰€πΆπΊ(𝐹(𝐺)). Since 𝐺 is solvable, it follows by [15, Theorem 2.6, page 216] that 𝐢𝐺(𝐹(𝐺))≀𝐹(𝐺)=𝑂𝑝(𝐺): a contradiction. Thus we must have 𝐺=π‘ƒπΊπ‘ž. Let πΊβˆ—π‘ž be a Sylow π‘ž-subgroup of 𝑁𝐺(𝑃). By (2), we have that πΊβˆ—π‘žβŠ²π‘πΊ(𝑃). Hence 𝑁𝐺(𝑃)=π‘ƒπΊβˆ—π‘ž=π‘ƒΓ—πΊβˆ—π‘ž. Thus 𝑃≀𝑍(𝑁𝐺(𝑃)), and, therefore, 𝐺 is 𝑝-nilpotent by [15, Theorem 4.3, page 252]: a final contradiction.

We need the following result.

Theorem 3.2. Let β„± be a saturated formation containing the class of supersolvable groups 𝒰. Let 𝐺𝑝 be a normal Sylow 𝑝-subgroup of a group 𝐺 such that 𝐺/πΊπ‘βˆˆβ„±. Fix an ALPE-subgroup 𝑃 of 𝐺𝑝 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then πΊβˆˆβ„±.

Proof. We treat the following two cases.
Case 1. 𝑂𝑝′(𝐺)β‰ 1.
Clearly, 𝐺𝑝𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) is a normal Sylow 𝑝-subgroup of 𝐺/𝑂𝑝′(𝐺) and 𝑃𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) is an ALPE-subgroup of 𝐺𝑝𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) having maximal order. By hypothesis and Lemma 2.1, the ALPE-subgroups of 𝑃𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺) are 𝑆-semipermutable in 𝐺/𝑂𝑝′(𝐺). Clearly, 𝐺/𝐺𝑝𝐺𝑝𝑂𝑝′(𝐺)/𝐺𝑝≅𝐺𝐺𝑝𝑂𝑝′≅(𝐺)𝐺/𝑂𝑝′(𝐺)𝐺𝑝𝑂𝑝′(𝐺)/𝑂𝑝′(𝐺)βˆˆβ„±.(3.1)
Thus, our hypothesis carries over to 𝐺/𝑂𝑝′(𝐺) and so 𝐺/𝑂𝑝′(𝐺)βˆˆβ„± by induction on the order of 𝐺. Therefore, 𝐺/(𝑂𝑝′(𝐺)βˆ©πΊπ‘)β‰…πΊβˆˆβ„±.

Case 2. 𝑂𝑝′(𝐺)=1.
Let 𝐻 be an ALPE-subgroup of 𝑃. Then 𝐻 is 𝑆-quasinormal in 𝐺 by Lemma 2.3 and hence 𝑂𝑝(𝐺)≀𝑁𝐺(𝐻) by Lemma 2.2. Let 𝑇=𝑃𝑂𝑝(𝐺). Then 𝐻 is normal in 𝑇. Thus Lemma 2.4 implies that 𝑃 is supersolvably embedded in 𝑇. Then, 𝑇/𝐢𝑇(𝑃) is supersolvable by [14, Lemma 7.15, page 35]. Clearly, 𝑇𝑝=πΊπ‘βˆ©π‘‡βŠ²π‘‡, where 𝑇𝑝 is a Sylow 𝑝-subgroup of 𝑇. Let 𝑄 be a π‘ξ…ž-subgroup of 𝐢𝑇(𝑃). Then 𝑄𝑃=𝑄×𝑃 is a group of automorphisms of 𝑇𝑝=𝑂𝑝(𝑇). But 𝐢𝑇𝑝(𝑃)=𝑃, and consequently, 𝑄 acts trivially on 𝐢𝑇𝑝(𝑃). Then 𝑄 acts trivially on 𝑇𝑝 by [15, Theorem 3.4, page 179], that is, 𝑄≀𝐢𝑇(𝑇𝑝). It is easy to see that 𝑇 is subnormal in 𝐺 and so 𝑂𝑝′(𝑇)≀𝑂𝑝′(𝐺)=1.  Hence 𝐹(𝑇)=𝑇𝑝.  Since 𝑇 is solvable, it follows by [15, Theorem 2.6, page 216] that 𝑄≀𝐢𝑇(𝐹(𝑇))≀𝐹(𝑇)=𝑇𝑝: a contradiction. Hence 𝐢𝑇(𝑃) must be a 𝑝-group and so 𝐢𝑇(𝑃)=𝑃. Thus, 𝑇/𝐢𝑇(𝑃)=𝑇/𝑃 is supersolvable which implies that 𝑇 is supersolvable by [16, Theorem 4]. Thus 𝑂𝑝(𝐺) is supersolvable and therefore, 𝐺=𝐺𝑝𝑂𝑝(𝐺) is supersolvable by [17, Exercise 7.2.23, page 159]. Hence, πΊβˆˆπ’°βŠ†β„±.

As an immediate consequence of Theorem 3.2, we have the following theorem.

Corollary 3.3. Let 𝐺𝑝 be a normal Sylow 𝑝-subgroup of a group 𝐺 such that 𝐺/𝐺𝑝 is supersolvable. Fix an ALPE-subgroup 𝑃 of 𝐺𝑝 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

We now prove the following theorem.

Theorem 3.4. Let 𝐺 be a group. For every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺𝑝 of 𝐺, fix an ALPE-subgroup 𝑃 of 𝐺𝑝 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

Proof. By repeated applications of Theorem 3.1, the group 𝐺 has a Sylow tower of supersolvable type. Hence 𝐺 has a normal Sylow 𝑝-subgroup 𝐺𝑝, where 𝑝 is the largest prime dividing the order of 𝐺. By Lemma 2.1, our hypothesis carries over to 𝐺/𝐺𝑝. Thus 𝐺/𝐺𝑝 is supersolvable by induction on the order of 𝐺. Now, it follows from Corollary 3.3 that 𝐺 is supersolvable.

As an immediate consequence of Theorem 3.4, we have the following corollary.

Corollary 3.5 (Asaad et al. [7]). If 𝐺 is a group such that the ALPE-subgroups of every Sylow subgroup of 𝐺 are normal in 𝐺, then 𝐺 is supersolvable.

Corollary 3.6 (Ramadan [8]). If 𝐺 is a group such that the ALPE-subgroups of every Sylow subgroup of 𝐺 are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable.

We need the following Lemma.

Lemma 3.7. Let 𝐾 be a normal 𝑝-subgroup of a group 𝐺 such that 𝐺/𝐾 is supersolvable. Fix an ALPE-subgroup 𝑃 of 𝐾 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

Proof. Let 𝐺𝑝 be a Sylow 𝑝-subgroup of 𝐺. We treat the following two cases.
Case 1. 𝐾=𝐺𝑝.
Then by Corollary 3.3, 𝐺 is supersolvable.

Case 2. 𝐾<𝐺𝑝.
Put πœ‹(𝐺)={𝑝1,𝑝2,…,𝑝𝑛}, where 𝑝1>𝑝2>β‹―>𝑝𝑛. Since 𝐺/𝐾 is supersolvable, it follows by [18, Theorem 5, page 5] that 𝐺/𝐾 possesses supersolvable subgroups 𝑀/𝐾 and 𝐿/𝐾 such that |𝐺/πΎβˆΆπ‘€/𝐾|=𝑝1 and |𝐺/𝐾∢𝐿/𝐾|=𝑝𝑛. Since 𝑀/𝐾 and 𝐿/𝐾 are supersolvable, it follows that 𝑀 and 𝐿 are supersolvable by induction on the order of 𝐺. Since |πΊβˆΆπ‘€|=|𝐺/πΎβˆΆπ‘€/𝐾|=𝑝1 and |𝐺∢𝐿|=|𝐺/𝐾∢𝐿/𝐾|=𝑝𝑛, it follows again by [18, Theorem 5, page 5] that 𝐺 is supersolvable.

Now, we can prove the following theorem.

Theorem 3.8. Let 𝐾 be a normal subgroup of 𝐺 such that 𝐺/𝐾 is supersolvable. For every prime 𝑝 dividing the order of 𝐾 and every Sylow 𝑝-subgroup 𝐾𝑝 of 𝐾, fix an ALPE-subgroup 𝑃 of 𝐾𝑝 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

Proof. By Lemma 2.1, the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐾. Hence 𝐾 is supersolvable by Theorem 3.4. Thus 𝐾 has a normal Sylow 𝑝-subgroup 𝐾𝑝, where 𝑝 is the largest prime dividing the order of 𝐾. Since 𝐾𝑝 is characteristic in 𝐾 and 𝐾⊲𝐺, we have that πΎπ‘βŠ²πΊ. Clearly, (𝐺/𝐾𝑝)/(𝐾/𝐾𝑝)≅𝐺/𝐾 is supersolvable. By Lemma 2.1, our hypothesis carries over to 𝐺/𝐾𝑝 and hence 𝐺/𝐾𝑝 is supersolvable by induction on the order of 𝐺. Now, it follows from Lemma 3.7 that 𝐺 is supersolvable.

As an immediate consequence of Theorem 3.8, we have the following corollary.

Corollary 3.9 (Ramadan [8]). Let 𝐾 be a normal subgroup of a group 𝐺 such that 𝐺/𝐾 is supersolvable. If the ALPE-subgroups of every Sylow subgroup of 𝐾 are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable.

4. Similar Results

Following similar arguments to those of Theorem 3.1, it is possible to prove the following result.

Theorem 4.1. Let 𝑝 be the smallest prime dividing the order of a group 𝐺 and let 𝐺𝑝 be a Sylow 𝑝-subgroup of 𝐺. Fix an ALPE-subgroup 𝑃 of Ξ©(𝐺𝑝) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is 𝑝-nilpotent.

We prove the following lemma.

Lemma 4.2. Let 𝐾 be a normal 𝑝-subgroup of a group 𝐺 such that 𝐺/𝐾 is supersolvable. Fix an ALPE-subgroup 𝑃 of Ξ©(𝐾) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

Proof. Let 𝐺𝑝 be a Sylow 𝑝-subgroup of 𝐺. We treat the following two cases.
Case 1 𝐾=𝐺𝑝. By [15, Theorem 2.1, page 221], there exists a π‘ξ…ž-Hall subgroup 𝑇, which is a complement to 𝐺𝑝 in 𝐺. Hence 𝐺/𝐺𝑝≅𝑇 is supersolvable. Since Ξ©(𝐺𝑝) is characteristic in 𝐺𝑝 and πΊπ‘βŠ²πΊ, we have that Ξ©(𝐺𝑝)⊲𝐺. Clearly, Ξ©(𝐺𝑝)𝑇/Ξ©(𝐺𝑝)≅𝑇 is supersolvable. Thus, our hypothesis and Corollary 3.3 imply that Ξ©(𝐺𝑝)𝑇 is supersolvable. Therefore, 𝐺 is supersolvable by Lemma 2.7.
Case 2 𝐾<𝐺. Put πœ‹(𝐺)={𝑝1,𝑝2,…,𝑝𝑛}, where 𝑝1>𝑝2>β‹―>𝑝𝑛. Since 𝐺/𝐾 is supersolvable, it follows by [18, Theorem 5, page 5] that 𝐺/𝐾 possesses supersolvable subgroups 𝑀/𝐾 and 𝐿/𝐾 such that |𝐺/πΎβˆΆπ‘€/𝐾|=𝑝1 and |𝐺/𝐾∢𝐿/𝐾|=𝑝𝑛. Since 𝑀/𝐾 and 𝐿/𝐾 are supersolvable, it follows that 𝑀 and 𝐿 are supersolvable by induction on the order of 𝐺. Since |πΊβˆΆπ‘€|=|𝐺/πΎβˆΆπ‘€/𝐾|=𝑝1 and |𝐺∢𝐿|=|𝐺/𝐾∢𝐿/𝐾|=𝑝𝑛, it follows again by [18, Theorem 5, page 5] that 𝐺 is supersolvable.

By a similar proof to the proof of Theorem 3.4, we can prove the following theorem.

Theorem 4.3. Let 𝐺 be a group. For every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺𝑝 of 𝐺,fix an ALPE-subgroup 𝑃 of Ξ©(𝐺𝑝) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

As an immediate consequence of Theorem 4.3, we have the following corollary.

Corollary 4.4 (Asaad et al. [7]). If 𝐺 is a group such that for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺𝑝, the ALPE-subgroups of Ξ©(𝐺𝑝) are normal in 𝐺, then 𝐺 is supersolvable.

Corollary 4.5 (Ramadan [8]). If 𝐺 is a group such that for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺𝑝, the ALPE-subgroups of Ξ©(𝐺𝑝) are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable.

We can now prove the following corollary.

Corollary 4.6. Let 𝐾 be a normal subgroup of 𝐺 such that 𝐺/𝐾 is supersolvable. For every prime 𝑝 dividing the order of 𝐾 and every Sylow 𝑝-subgroup 𝐾𝑝 of 𝐾, fix an ALPE-subgroup 𝑃 of Ξ©(𝐾𝑝) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

Proof. By Lemma 2.1, the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐾. Hence 𝐾 is supersolvable by Theorem 4.3. Thus 𝐾 has a normal Sylow 𝑝-subgroup 𝐾𝑝, where 𝑝 is the largest prime dividing the order of 𝐾. Since 𝐾𝑝 is characteristic in 𝐾 and 𝐾⊲𝐺, we have that πΎπ‘βŠ²πΊ. Clearly, (𝐺/𝐾𝑝)/(𝐾/𝐾𝑝)≅𝐺/𝐾 is supersolvable. By Lemma 2.1, the hypothesis of our theorem carries over to 𝐺/𝐾𝑝. Thus 𝐺/𝐾𝑝 is supersolvable by induction on the order of 𝐺 and it follows that 𝐺 is supersolvable by Lemma 4.2.

Remarks 4.7. (a) The converse of Theorem 3.4 is not true. For example, set 𝐺=𝑆3×𝑍3, where 𝑆3=⟨π‘₯,π‘¦βˆ£π‘₯3=𝑦2=1,𝑦π‘₯=π‘₯2π‘¦βŸ© and 𝑍3=βŸ¨π‘§βˆ£π‘§3=1⟩. Clearly, 𝐺 is supersolvable and 𝐺 has an abelian Sylow 3-subgroup of exponent 3. It is easy to check that 𝐺 contains a subgroup ⟨π‘₯π‘§βŸ© of order 3 which fails to be 𝑆-semipermutable in 𝐺.
(b) Theorem 4.3 is not true when the smallest prime dividing the order of 𝐺 is even and Ξ©(𝐺𝑝)=Ξ©1(𝐺𝑝), where 𝐺𝑝 is a Sylow 𝑝-subgroup of 𝐺. For example, if 𝑄 is the quaternion group βŸ¨π‘Ž,π‘βˆ£π‘Ž4=1,𝑏2=π‘Ž2,π‘βˆ’1π‘Žπ‘=π‘Žβˆ’1⟩, 𝐢9 is a cyclic group of order 9 with generator 𝑐, and the action of 𝐢9 on 𝑄 is given by π‘Žπ‘=𝑏,𝑏𝑐=π‘Žπ‘, then the semidirect product of 𝑄 by 𝐢9 is a group of even order in which every subgroup of prime order is 𝑆-semipermutable. Clearly, the semidirect product of 𝑄 by 𝐢9 is not supersolvable (see Buckley [4, Examples (ii)]).

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