Recognition of \(A_1\)\(_0\) and \(L_4\)(4) by two special conjugacy class sizes

Abstract

It is well-known that A₁₀ is the smallest (by order) nonabelian simple group with connected prime graph and L₄(4) is the smallest nonabelian simple group of Lie type with connected prime graph.  In 2009, A.V. Vasil'ev first dealt with the groups with connected prime graph and proved that Thompson's conjecture holds for A₁₀ and L₄(4) (see [1]). In this work, the authors characterize finite simple groups A₁₀ and L₄(4) by their orders and largest and smallest conjugacy class sizes greater than 1, and partially generalize A.V. Vasil'ev's work.