On products of groups and indices not divisible by a given prime

Abstract

Let the group G = AB be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each pregular element of prime power order x ∈ A ∪ B if and only if G is p-decomposable, i.e. G = O p (G) × O p (G).