On Ditsman’s lemma

Abstract

Let H be a subgroup of a group G generated by a finite G-invariant subset X = tU i=1 k C i that consists of elements of finite order, where C i is a class of conjugate elements of G with representative a i . We prove that $$|H| \leqslant \prod\limits_{i = 1}^k {o(a_i )^{|C_i |} } ,$$ where o(a i ) is the order of the element a i ∈ C i . Best estimates are obtained for some important special cases.