On the product of two $\pi$-decomposable groups

Abstract

The aim of this paper is to prove the following result: Let π be a set of odd primes. If the finite group G = AB is a product of two π-decomposable subgroups A = Oπ(A) × O π (A) and B = Oπ(B) × O π (B), then Oπ(A)Oπ(B) = Oπ(B)Oπ(A) and this is a Hall π-subgroup of G.

The $D_π$-property on products of $π$-decomposable groups

“…(9) Assume finally that N is a simple group of Lie type of characteristic p. If P is any Sylow p-subgroup of N , then C Aut(N ) (P ) is a p-group, so it follows from (6) that p ∈ π ′ . Now, if N has Lie rank l > 1, arguing as in the proof of [18,Lemma 12], we can deduce that p ∈ π(B ∩ N ).…”

“…Since p ∈ π(N ∩ B) by 1 (9) this forces that A ≤ X when p = 3. For the case p = 3, we get that [18,Lemma 6], and this also implies that A ≤ X. Since q 3 − 1 divides |G : Y | we get that r = q 3 ∈ π(A).…”

“…From [2, Lemma 2.5] the order of a soluble subgroup of N whose order is divisible by r should divide either m(q m − 1), or 2m 2 (m + 1)(q − 1)l with q = p, r = m + 1, m = 2 l . But in this last case, r = m + 1 is the only primitive prime divisor of such group with respect to the pair (q, m) and so, applying [21, 2.4 Proposition D] it should be (q, m) ∈ {(2, 4), (2,10), (2,12), (2,18), (3,4), (3,6), (5,6)}, but this contradicts the fact that m is a power of 2 and |π(G)| ≥ 5. Hence A ∩ N is a soluble subgroup of order dividing m(q m − 1).…”

“…In fact the results in [14] make up the starting point of a longtime development carried out in [14,15,16,17,18], where the existence of Hall π-subgroups have been considered as a preliminary step to finding conditions of π-separability for products of π-decomposable subgroups. The goal was to prove the following theorem, which was first stated as a conjecture in [15]: Theorem 1.…”

Finite Trifactorised Groups and -Decomposability

“…This theorem, whose proof uses the classification of finite simple groups, is part of a development carried out in [8,9,11,12] and motivated by the search for extensions of the theorem of Kegel and Wielandt mentioned above (see also [10]). We apply Theorem 1.1 to obtain new results on trifactorised groups within the general universe of finite groups.…”