Finite Trifactorized Groups and Formations

Finite Trifactorised Groups and -Decomposability

“…Since G F = C F H is an elementary abelian subgroup by Step 5, again by Maschke's theorem, there exists a complement of H in G F , say T , which is G σ -invariant. But then, by Step 6, we see that As a particular case of Theorem 3.11 we recover the following extension of Kegel's result quoted in the introduction, which appears in [3]. Corollary 3.13.…”

“…A much deeper result in the universe of all finite groups was proved by Kazarin in [7] using the classification of finite simple groups: if the group G = AB = AC = BC is the product of three soluble subgroups A, B and C, then G is soluble. Some related results were obtained in [3], again in the universe of soluble groups, by considering some wellknown families of subgroup-closed saturated formations of so-called nilpotent type (see [5] for an account of such classes of groups).…”

Nilpotent-like fitting formations of finite soluble groups

“…is a subgroup-closed Fitting class and F is integrated, it follows that F(p) C S,,^) n T, [3] Finite soluble groups 429 for every prime p . Assume that there exists a prime p such that…”