| Title:
|
Characterization of the alternating groups by their order and one conjugacy class length (English) |
| Author:
|
Khalili Asboei, Alireza |
| Author:
|
Mohammadyari, Reza |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
63-70 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a finite group, and let $N(G)$ be the set of conjugacy class sizes of $G$. By Thompson's conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N(G)=N(L)$, then $L $ and $G$ are isomorphic. Recently, Chen et al.\ contributed interestingly to Thompson's conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li's PhD dissertation). In this article, we investigate validity of Thompson's conjecture under a weak condition for the alternating groups of degrees $p+1$ and $p+2$, where $p$ is a prime number. This work implies that Thompson's conjecture holds for the alternating groups of degree $p+1$ and $p+2$. (English) |
| Keyword:
|
finite simple group |
| Keyword:
|
conjugacy class size |
| Keyword:
|
prime graph |
| Keyword:
|
Thompson's conjecture |
| MSC:
|
20D06 |
| MSC:
|
20D08 |
| MSC:
|
20D60 |
| idZBL:
|
Zbl 06587873 |
| idMR:
|
MR3483222 |
| DOI:
|
10.1007/s10587-016-0239-0 |
| . |
| Date available:
|
2016-04-07T14:54:15Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144875 |
| . |
| Reference:
|
[1] Asboei, A. K., Mohammadyari, R.: Recognizing alternating groups by their order and one conjugacy class length.J. Algebra. Appl. 15 (2016), 7 pages \hfil doi: 10.1142/S0219498816500213. Zbl 1336.20026, MR 3405720, 10.1142/S0219498816500213 |
| Reference:
|
[2] Chen, G.: Further reflections on Thompson's conjecture.J. Algebra 218 (1999), 276-285. MR 1704687, 10.1006/jabr.1998.7839 |
| Reference:
|
[3] Chen, G.: On Thompson's conjecture.J. Algebra 185 (1996), 184-193. MR 1409982, 10.1006/jabr.1996.0320 |
| Reference:
|
[4] Chen, G. Y.: On Thompson's Conjecture.PhD thesis Sichuan University, Chengdu (1994). |
| Reference:
|
[5] Chen, Y., Chen, G.: Recognizing $L_{2}(p)$ by its order and one special conjugacy class size.J. Inequal. Appl. (electronic only) (2012), 2012 Article ID 310, 10 pages. Zbl 1282.20016, MR 3027693 |
| Reference:
|
[6] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups.Clarendon Press, Oxford (1985). Zbl 0568.20001, MR 0827219 |
| Reference:
|
[7] Gorenstein, D.: Finite Groups.Chelsea Publishing Company, New York (1980). Zbl 0463.20012, MR 0569209 |
| Reference:
|
[8] Hagie, M.: The prime graph of a sporadic simple group.Comm. Algebra 31 (2003), 4405-4424. Zbl 1031.20009, MR 1995543, 10.1081/AGB-120022800 |
| Reference:
|
[9] Iiyori, N., Yamaki, H.: Prime graph components of the simple groups of Lie type over the field of even characteristic.J. Algebra 155 (1993), 335-343. MR 1212233, 10.1006/jabr.1993.1048 |
| Reference:
|
[10] Iranmanesh, A., Alavi, S. H., Khosravi, B.: A characterization of $PSL(3,q)$ where $q$ is an odd prime power.J. Pure Appl. Algebra 170 (2002), 243-254. Zbl 1001.20005, MR 1904845, 10.1016/S0022-4049(01)00113-X |
| Reference:
|
[11] Iranmanesh, A., Khosravi, B.: A characterization of $F_{4}(q)$ where $q$ is an odd prime power.London Math. Soc. Lecture Note Ser. 304 (2003), 277-283. Zbl 1058.20016, MR 2051533 |
| Reference:
|
[12] Iranmanesh, A., Khosravi, B., Alavi, S. H.: A characterization of PSU$_{3}(q)$ for $q>5$.Southeast Asian Bull. Math. 26 (2002), 33-44. Zbl 1019.20012, MR 2046570 |
| Reference:
|
[13] Kondtratev, A. S., Mazurov, V. D.: Recognition of alternating groups of prime degree from their element orders.Sib. Math. J. 41 (2000), 294-302. MR 1762188, 10.1007/BF02674599 |
| Reference:
|
[14] Li, J. B.: Finite Groups with Special Conjugacy Class Sizes or Generalized Permutable Subgroups.PhD thesis Southwest University, Chongqing (2012). |
| Reference:
|
[15] Mazurov, V. D., Khukhro, E. I.: The Kourovka Notebook. Unsolved Problems in Group Theory. Including Archive of Solved Problems.Institute of Mathematics, Russian Academy of Sciences Siberian Div., Novosibirsk (2006). Zbl 1084.20001, MR 2263886 |
| Reference:
|
[16] Vasil'ev, A.: On connection between the structure of a finite group and the properties of its prime graph.Sib. Math. J. 46 (2005), 396-404. Zbl 1096.20019, MR 2164556, 10.1007/s11202-005-0042-x |
| Reference:
|
[17] Williams, J. S.: Prime graph components of finite groups.J. Algebra. 69 (1981), 487-513. Zbl 0471.20013, MR 0617092, 10.1016/0021-8693(81)90218-0 |
| . |
