Number of Distinct Homomorphic Images in Coset Diagrams

<div>Homomorphic images <span class="nowrap"><svg height="18.5962pt" id="M1158" style="vertical-align:-7.004801pt" version="1.1" viewbox="-0.0498162 -11.5914 40.7727 18.5962" width="40.7727pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.841,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.273,-5.741)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.841,3.784)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.099,3.784)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,15.337,5.885)"><path d="M443 227C443 310 376 399 246 399C216 399 175 385 139 374L163 546H394L422 615L414 623H124L84 318C106 325 144 336 186 336C298 336 353 275 353 189C353 120 311 45 229 45C168 45 125 77 103 97C92 105 84 102 73 94C61 84 52 69 51 61C48 47 52 40 66 25C84 9 127 -12 175 -12C229 -11 296 15 350 59C413 107 443 167 443 227Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,19.059,3.784)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,21.215,3.784)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,23.454,5.885)"><path d="M318 -137C221 -59 159 100 159 272S220 600 318 680L294 710C146 612 80 452 80 273V272C80 91 146 -72 294 -168L318 -137Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,25.859,5.885)"><path d="M461 212C461 317 386 401 277 401C257 401 234 394 212 384L145 351C175 478 268 537 333 566C371 583 409 595 443 601L435 641C396 634 343 622 305 610C193 570 37 459 37 240C37 88 127 -12 248 -12C372 -12 461 89 461 212ZM371 176C371 111 344 32 268 32C184 32 132 126 132 248C132 270 133 293 138 310C161 323 197 339 236 339C328 339 371 258 371 176Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,29.07,5.885)"><path d="M202 22C202 97 141 138 100 138C74 138 46 121 46 92C46 74 56 66 61 64C97 55 127 31 127 -5C127 -43 97 -68 44 -90L59 -129C127 -110 202 -64 202 22Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,30.669,5.885)"><path d="M443 227C443 310 376 399 246 399C216 399 175 385 139 374L163 546H394L422 615L414 623H124L84 318C106 325 144 336 186 336C298 336 353 275 353 189C353 120 311 45 229 45C168 45 125 77 103 97C92 105 84 102 73 94C61 84 52 69 51 61C48 47 52 40 66 25C84 9 127 -12 175 -12C229 -11 296 15 350 59C413 107 443 167 443 227Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,33.88,5.885)"><path d="M291 272C291 452 228 611 76 710L51 680C147 602 211 444 211 272S151 -55 51 -137L76 -168C227 -66 291 91 291 272Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,36.831,3.784)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"></path></g></svg>.</span></div>

Journal of Mathematics

Number of Distinct Homomorphic Images in Coset Diagrams

Figure 38