Central European Journal of Mathematics

, Volume 10, Issue 2, pp 722–739

# Schreier type theorems for bicrossed products

• Ana Agore
• Gigel Militaru
Research Article

## 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; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: GH in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H α⋈β GH α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H α⋈β GH α′⋈β′ 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 products of groups are given.

## Keywords

Matched pairs Bicrossed product of groups

## MSC

20B05 20B35 20D06 20D40

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## Authors and Affiliations

1. 1.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania