Knit Products of Some Groups and Their Applications

Abstract

Let G be a group with subgroups A and K (not necessarily normal) such that G = AK and A \cap K = \{1\} . Then G is isomorphic to the knit product , that is, the “two-sided semidirect product” of K by A . We note that knit products coincide with Zappa-Szep products (see [18]). In this paper, as an application of [2, Lemma 3.16], we first define a presentation for the knit product G where A and K are finite cyclic subgroups. Then we give an example of this presentation by considering the (extended) Hecke groups. After that, by defining the Schur multiplier of G , we present sufficient conditions for the presentation of G to be efficient. In the final part of this paper, by examining the knit product of a free group of rank n by an infinite cyclic group, we give necessary and sufficient conditions for this special knit product to be subgroup separable.