Finite Trifactorised Groups and -Decomposability

Abstract

We derive some structural properties of a trifactorised finite group $G=AB=AC=BC$, where $A$, $B$, and $C$ are subgroups of $G$, provided that $A=A_{\unicode[STIX]{x1D70B}}\times A_{\unicode[STIX]{x1D70B}^{\prime }}$ and $B=B_{\unicode[STIX]{x1D70B}}\times B_{\unicode[STIX]{x1D70B}^{\prime }}$ are $\unicode[STIX]{x1D70B}$-decomposable groups, for a set of primes $\unicode[STIX]{x1D70B}$.

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

“…and A∩ B ≤ S.) In this setting, in [19] we proved the following result, which can be considered as a particular significant case of our Main Theorem:…”

“…In fact, our study on the D π -property was initially motivated by the development carried out in [19] on trifactorized groups and π-decomposability, where the dominance property appeared as a relevant tool. Trifactorized groups, that is, groups of the form G = AB = AC = BC, where A, B, and C are subgroups of G, play a key role within the study of factorized groups.…”

“…([19, Theorem 3.2]) Let π be a set of odd primes. Let the group G = AB = AC = BC be the product of three subgroups A, B and C, where…”

The $$D_{\pi }$$-property on products of $$\pi $$-decomposable groups

“…Specifically, for a subgroup N of a group G = AB which is the product of subgroups A and B, the factorizer of N in G, denoted X(N ), is the intersection of all factorized subgroups of G containing N . Recall that a subgroup S of G = AB is factorized if S = (S ∩ A)(S ∩ B) and A ∩ B ≤ S. In this setting, in [22] we proved the following result, which can be considered as a particular significant case of our Main Theorem: Then G is a Dπ-group.…”

“…In fact, our study on the Dπ-property was initially motivated by the development carried out in [22] on trifactorized groups and π-decomposability, where the dominance property appeared as a relevant tool. Trifactorized groups, that is, groups of the form G = AB = AC = BC, where A, B, and C are subgroups of G, play a key role within the study of factorized groups.…”