On the product of a π-group and a π-decomposable group

Abstract

The main result in the paper states the following: Let π be a set of odd primes. Let the finite group G = AB be the product of a π -decomposable subgroup A = O π (A) × O π (A) and a π -subgroup B. Then O π (A) O π (G); equivalently the group G possesses Hall π -subgroups. In this case O π (A)B is a Hall π -subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G.

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

“…More precisely we complete the study on products of π-decomposable groups carried out in [14] and [15] (see also [17]) and prove the following general result:…”

“…As approaches to the aimed result, we presented in [14] and [15] several particular positive cases, namely, when either one of the factors is a π-group ([14, Theorem 1, Lemma 1]), or they are soluble groups ([15, Theorem 2]), or when the factors have coprime orders ([15, Proposition 1]). These results largely extend the above mentioned known ones on products of 2-decomposable groups.…”

On the product of two π-decomposable soluble groups

“…It is worth recalling the following result, which is Lemma 1 in [12] and provides an equivalent statement to this theorem. Lemma 1.…”

“…More precisely we take further the study that was started in [12]. The main result in that paper states the following: Theorem 1.…”

“…In [12,Final examples,3] it has been shown that the conclusion of Theorem 3 is not true for the groups L 2 (2 n ), n ≥ 2, except if either n = 3 or 2 n + 1 is a Fermat prime.…”