On σ-subnormal subgroups of factorised finite groups

Let sigma = {sigma(i) : i is an element of I} be a partition of the set P of all prime numbers. A subgroup X of a finite group G is called sigma-subnormal in G if there is chain of subgroups X = X-0 subset of X-1 subset of ... subset of X-n = G with Xi-1 normal in X-i or X-i/Core(Xi)(Xi-1) is a a sigma(i)-group for some i is an element of I, 1 <= i <= n. In the special case that sigma is the partition of P into sets containing exactly one prime each, the sigma-subnormality reduces to the familiar case of subnormality. If a finite soluble group G = AB is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is sigma-subnormal in < X, X-g > for all g is an element of A boolean OR B, we prove that X is sigma-subnormal in G. This is an extension of a subnormality criteria due to Maier and Sidki and Casolo. (C) 2020 Elsevier Inc. All rights reserved.