For a finite group G, we write to denote the prime divisor set of the various conjugacy class lengths of G and the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that can be bounded by . Let G be an almost simple group G such that the graph built on element orders is a tree. By using Lucido’s classification theorem, we prove except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with a forest, the inequality is true.
Published in | Mathematics and Computer Science (Volume 1, Issue 1) |
DOI | 10.11648/j.mcs.20160101.14 |
Page(s) | 17-20 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2016. Published by Science Publishing Group |
Prime Graph, Conjugacy Class Length, Almost Simple Group
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APA Style
Liguo He, Yaping Liu, Jianwei Lu. (2016). Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree. Mathematics and Computer Science, 1(1), 17-20. https://doi.org/10.11648/j.mcs.20160101.14
ACS Style
Liguo He; Yaping Liu; Jianwei Lu. Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree. Math. Comput. Sci. 2016, 1(1), 17-20. doi: 10.11648/j.mcs.20160101.14
@article{10.11648/j.mcs.20160101.14, author = {Liguo He and Yaping Liu and Jianwei Lu}, title = {Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree}, journal = {Mathematics and Computer Science}, volume = {1}, number = {1}, pages = {17-20}, doi = {10.11648/j.mcs.20160101.14}, url = {https://doi.org/10.11648/j.mcs.20160101.14}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20160101.14}, abstract = {For a finite group G, we write to denote the prime divisor set of the various conjugacy class lengths of G and the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that can be bounded by . Let G be an almost simple group G such that the graph built on element orders is a tree. By using Lucido’s classification theorem, we prove except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with a forest, the inequality is true.}, year = {2016} }
TY - JOUR T1 - Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree AU - Liguo He AU - Yaping Liu AU - Jianwei Lu Y1 - 2016/05/28 PY - 2016 N1 - https://doi.org/10.11648/j.mcs.20160101.14 DO - 10.11648/j.mcs.20160101.14 T2 - Mathematics and Computer Science JF - Mathematics and Computer Science JO - Mathematics and Computer Science SP - 17 EP - 20 PB - Science Publishing Group SN - 2575-6028 UR - https://doi.org/10.11648/j.mcs.20160101.14 AB - For a finite group G, we write to denote the prime divisor set of the various conjugacy class lengths of G and the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that can be bounded by . Let G be an almost simple group G such that the graph built on element orders is a tree. By using Lucido’s classification theorem, we prove except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with a forest, the inequality is true. VL - 1 IS - 1 ER -