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.