Abstract
In the past thirty years, many authors investigated some quantitative characterizations of finite groups, especially finite simple groups, such as quantitative characterizations by group order and element orders, by the set of lengths of conjugacy classes, by dimensions of irreducible characters, etc. In this article the projective special linear group is characterized by its order and one special conjugacy class size, where p is a prime. This work implies that Thompson’s conjecture holds for . MSC:20D08, 20D60.
Highlights
All groups considered in this paper are finite, and simple groups are non-Abelian
In, Chen proved in his Ph.D. dissertation [ ] that Thompson’s conjecture holds for all simple groups with a non-connected prime graph
Every minimal normal subgroup of G is non-solvable, as desired. It follows that M ≤ G ≤ Aut(M), where M = S × · · · × Sk and Si is a direct product of isomorphic non-Abelian simple groups for i =, . . . , k
Summary
All groups considered in this paper are finite, and simple groups are non-Abelian. For convenience, we use π(n) and np to denote the set of prime divisors and p-part of the nature number n, respectively. In , Chen proved in his Ph.D. dissertation [ ] that Thompson’s conjecture holds for all simple groups with a non-connected prime graph In , Vasil’ev first dealt with the simple groups with a connected prime graph and proved that Thompson’s conjecture holds for A and L ( ) (see [ ]). We characterize the projective special linear group L (p) by its order and one special conjugacy class length, where p is a prime. Every minimal normal subgroup of G is non-solvable, as desired It follows that M ≤ G ≤ Aut(M), where M = S × · · · × Sk and Si is a direct product of isomorphic non-Abelian simple groups for i = , , . Proof Since the necessity of the theorem can be checked we only need to prove the sufficiency
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