On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups

Abstract

The generalized Fitting height h∗(G) of a finite group G is the least number h such that F∗ h(G) = G, where F∗ (0)(G) = 1, and F∗ (i+1)(G) is the inverse image of the generalized Fitting subgroup F∗(G/F∗ (i)(G)). Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series in which for i odd the factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor Gi+1/Gi is a (non-empty) direct product of nonabelian simple groups. Then h = λp(G) is called the non-p-soluble length of a group G. We proved that if a finite group G is a mutually permutable product of of subgroups A and B then max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1 and max{λp(A), λp(B)} = λp(G). Also we introduced and studied the non-Frattini length.