On the product of two π-decomposable soluble groups

Abstract

Let the group G = AB be a product of two π-decomposable sub-where π is a set of primes. The authors conjecture thatif π is a set of odd primes. In this paper it is proved that the conjecture is true if A and B are soluble. A similar result with certain additional restrictions holds in the case 2 ∈ π. Moreover, it is shown that the conjecture holds if O π (A) and O π (B) have coprime orders.

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

“…The next example, which appears in [14], shows that analogous results do not hold in general if the set of primes π contains the prime 2, although some related positive results were obtained in this case in [15]. Other examples in [14] and [15] give insight into occurring phenomena.…”

“…This result was announced as a conjecture in [15], [16] and [17], and also was mentioned in [4]. 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. Moreover, in [14] and [15] we obtained also some π-separability criteria for products of π-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.…”