# Schreier type theorems for bicrossed products

Open Mathematics (2012)

• Volume: 10, Issue: 2, page 722-739
• ISSN: 2391-5455

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## Abstract

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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: G → H 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 G ≅H α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H α⋈β G ≅ H 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.

## How to cite

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Ana Agore, and Gigel Militaru. "Schreier type theorems for bicrossed products." Open Mathematics 10.2 (2012): 722-739. <http://eudml.org/doc/269386>.

@article{AnaAgore2012,
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: G → H 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 G ≅H α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H α⋈β G ≅ H 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.},
author = {Ana Agore, Gigel Militaru},
journal = {Open Mathematics},
keywords = {Matched pairs; Bicrossed product of groups; matched pairs of groups; bicrossed products of groups; Hopf algebras; semidirect products},
language = {eng},
number = {2},
pages = {722-739},
title = {Schreier type theorems for bicrossed products},
url = {http://eudml.org/doc/269386},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Ana Agore
AU - Gigel Militaru
TI - Schreier type theorems for bicrossed products
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 722
EP - 739
AB - 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: G → H 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 G ≅H α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H α⋈β G ≅ H 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.
LA - eng
KW - Matched pairs; Bicrossed product of groups; matched pairs of groups; bicrossed products of groups; Hopf algebras; semidirect products
UR - http://eudml.org/doc/269386
ER -

## References

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