On the Intersection of Subgroup Products with Non-Cyclic Abelian Normal 2-Subgroups

Abstract

Sufficient conditions are established to ensure that the intersection of a product of two subgroups with an abelian normal subgroup is itself a subgroup within a finite group framework. It is proven that if K is an abelian 2-group and A is a subgroup of odd order, then the commutativity condition is sufficient to guarantee that the intersection  is a subgroup, regardless of whether K is cyclic or non-cyclic. This result generalizes and extends known findings from the cyclic case to the more general non-cyclic abelian setting.