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.

Key words： 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 (in Chinese)

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, WilsonR A. An Atlas of Finite Groups. Oxford: Clarendon Press, 1985

6 James G D, Kerber A. The Representation Theory of Symmetric Group. London: Addison Wesley Publishing Company, Inc, 1981, 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. 16th ed. Novosibirsk: Sobolev Institute of Mathematics, 2006

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, http://www.gap-system.org

11 Vasil’ev 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