Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei; Reza Mohammadyari

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 63-70
  • ISSN: 0011-4642

Abstract

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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 .

How to cite

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Khalili Asboei, Alireza, and Mohammadyari, Reza. "Characterization of the alternating groups by their order and one conjugacy class length." Czechoslovak Mathematical Journal 66.1 (2016): 63-70. <http://eudml.org/doc/276751>.

@article{KhaliliAsboei2016,
abstract = {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 $p+2$.},
author = {Khalili Asboei, Alireza, Mohammadyari, Reza},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite simple group; conjugacy class size; prime graph; Thompson's conjecture},
language = {eng},
number = {1},
pages = {63-70},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Characterization of the alternating groups by their order and one conjugacy class length},
url = {http://eudml.org/doc/276751},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Khalili Asboei, Alireza
AU - Mohammadyari, Reza
TI - Characterization of the alternating groups by their order and one conjugacy class length
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 63
EP - 70
AB - 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 $p+2$.
LA - eng
KW - finite simple group; conjugacy class size; prime graph; Thompson's conjecture
UR - http://eudml.org/doc/276751
ER -

References

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  1. Asboei, A. K., Mohammadyari, R., 10.1142/S0219498816500213, J. Algebra. Appl. 15 (2016), 7 pages doi: 10.1142/S0219498816500213. (2016) Zbl1336.20026MR3405720DOI10.1142/S0219498816500213
  2. Chen, G., 10.1006/jabr.1998.7839, J. Algebra 218 (1999), 276-285. (1999) MR1704687DOI10.1006/jabr.1998.7839
  3. Chen, G., 10.1006/jabr.1996.0320, J. Algebra 185 (1996), 184-193. (1996) MR1409982DOI10.1006/jabr.1996.0320
  4. Chen, G. Y., On Thompson's Conjecture, PhD thesis Sichuan University, Chengdu (1994). (1994) 
  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. (2012) Zbl1282.20016MR3027693
  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). (1985) Zbl0568.20001MR0827219
  7. Gorenstein, D., Finite Groups, Chelsea Publishing Company, New York (1980). (1980) Zbl0463.20012MR0569209
  8. Hagie, M., 10.1081/AGB-120022800, Comm. Algebra 31 (2003), 4405-4424. (2003) Zbl1031.20009MR1995543DOI10.1081/AGB-120022800
  9. Iiyori, N., Yamaki, H., 10.1006/jabr.1993.1048, J. Algebra 155 (1993), 335-343. (1993) MR1212233DOI10.1006/jabr.1993.1048
  10. Iranmanesh, A., Alavi, S. H., Khosravi, B., 10.1016/S0022-4049(01)00113-X, J. Pure Appl. Algebra 170 (2002), 243-254. (2002) Zbl1001.20005MR1904845DOI10.1016/S0022-4049(01)00113-X
  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. (2003) Zbl1058.20016MR2051533
  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. (2002) Zbl1019.20012MR2046570
  13. Kondtratev, A. S., Mazurov, V. D., 10.1007/BF02674599, Sib. Math. J. 41 (2000), 294-302. (2000) MR1762188DOI10.1007/BF02674599
  14. Li, J. B., Finite Groups with Special Conjugacy Class Sizes or Generalized Permutable Subgroups, PhD thesis Southwest University, Chongqing (2012). (2012) 
  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). (2006) Zbl1084.20001MR2263886
  16. Vasil'ev, A., 10.1007/s11202-005-0042-x, Sib. Math. J. 46 (2005), 396-404. (2005) Zbl1096.20019MR2164556DOI10.1007/s11202-005-0042-x
  17. Williams, J. S., 10.1016/0021-8693(81)90218-0, J. Algebra. 69 (1981), 487-513. (1981) Zbl0471.20013MR0617092DOI10.1016/0021-8693(81)90218-0

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