Schreier type theorems for bicrossed products

TL;DR

This paper develops a deformation method for matched pairs of groups to classify bicrossed products, leading to Schreier-type theorems that describe their structure and isomorphisms.

Contribution

It introduces a novel deformation approach for matched pairs of groups, enabling classification and explicit description of bicrossed product isomorphisms.

Findings

A deformation method for matched pairs of groups is established.

Schreier-type classification theorems for bicrossed products are proved.

A unique correspondence between isomorphisms and deformations is demonstrated.

Abstract

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), \alpha', \beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism $\sigma \in \Aut(H)$ then any $\sigma$-invariant isomorphism $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.