Nonsoluble length of finite groups with restrictions on Sylow subgroups

Abstract

ABSTRACTBy 𝔛(n) we denote the variety of all groups satisfying the law [x,y]n≡1, that is, groups with commutators of order dividing n. Let p be a prime and G a finite group whose Sylow p-subgroups have normal series of length k all of whose quotients belong to 𝔛(n). We show that the non-p-soluble length λp(G) of G is bounded in terms of k and n only (Theorem 1.2). In the case where p is odd, a stronger result is obtained (Theorem 1.3).