Schreier type theorems for bicrossed products

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 $(H,(G,\ast ),{\alpha }^{\prime },{\beta }^{\prime })$ such that there exist an $\sigma$-invariant isomorphism of groups $H{}_{\alpha }{\bowtie }_{\beta }G\cong H{}_{{\alpha }^{\prime }}{\bowtie }_{{\beta }^{\prime }}(G,\ast )$. Moreover, if we fix the group $H$ and the automorphism $σ\in \Aut(H)$ then any $\sigma$-invariant isomorphism $H{}_{\alpha }{\bowtie }_{\beta }G\cong H{}_{{\alpha }^{\prime }}{\bowtie }_{{\beta }^{\prime }}{G}^{\prime }$ 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.

arXiv comment21 pages, final version to appear in Central European J. Math