Thompson’s conjecture for alternating group of degree 22

Abstract

For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A₂₂ with connected prime graph.

Keywords

Finite group / conjugacy class size / simple group / prime graph of a group

References

[1.] Bi J X. A quantitative property of the length of the conjugacy classes of finite simple groups. J Liaoning Univ, 2008, 35: 5-6

[2.] Chen G Y. On Thompson’s conjecture. J Algebra, 1996, 185: 185-193

[3.] Chen G Y. Further reflections on Thompson’s conjecture. J Algebra, 1999, 218: 276-285

[4.] Chillag D, Herzog M. On the length of the conjugacy classes of finite groups. J Algebra, 1990, 131: 110-125

[5.] Conway J H, Curtis R T, Norton S P, Parker R A, Wilson R A. An Atlas of Finite Groups, 1985, Oxford: Clarendon Press

[6.] James G D, Kerber A. The Representation Theory of Symmetric Group, 1981, London: Addison Wesley Publishing Company, Inc 8 15

[7.] Khosravi A, Khosravi B. A new characterization of some alternating and symmetric groups. IJMMS, 2003, 45: 2863-2872

[8.] Khukhro E I, Mazurov V D. Unsolved Problems in Group Theory: the Kourovka Notebook, 2006 16th ed. Novosibirsk: Sobolev Institute of Mathematics

[9.] Kondrat’ev A S. On prime graph components of finite groups. Mat Sb, 1989, 180: 787-797

[10.] The GAP Group GAP-Groups, Algorithms, and Programming. Version 4.4.10, 2007

[11.] Vasilév A V. On Thompson’s conjecture. Siberian Electronic Mathematical Reports, 2009, 6: 457-464

[12.] Williams J S. Prime graph components of finite groups. J Algebra, 1981, 69: 489-513

[13.] Zavarnitsine A V. Finite simple groups with narrow prime spectrum. Siberian Electronic Mathematical Reports, 2009, 6: 1-12