Algebras and Representation Theory

, Volume 17, Issue 3, pp 831–848 | Cite as

Extending Structures I: The Level of Groups

Article

Abstract

Let H be a group and E a set such that \(H \subseteq E\). We shall describe and classify up to an isomorphism of groups that stabilizes H the set of all group structures that can be defined on E such that H is a subgroup of E. A general product, which we call the unified product, is constructed such that both the crossed product and the bicrossed product of two groups are special cases of it. It is associated to H and to a system \(\left( (S, 1_S, \ast), \triangleleft, \, \triangleright, \, f \right)\) called a group extending structure and we denote it by \(H \ltimes S\). There exists a group structure on E containing H as a subgroup if and only if there exists an isomorphism of groups \((E, \cdot) \cong H \ltimes S\), for some group extending structure \(\left( (S, 1_S, \ast), \triangleleft, \, \triangleright, \, f \right)\). All such group structures on E are classified up to an isomorphism of groups that stabilizes H by a cohomological type set \({\mathcal K}^{2}_{\ltimes} (H, (S, 1_S))\). A Schreier type theorem is proved and an explicit example is given: it classifies up to an isomorphism that stabilizes H all groups that contain H as a subgroup of index 2.

Keywords

The extension problem (bi)Crossed product Knit (Zappa–Szep) product 

Mathematics Subject Classifications (2010)

20A05 20E22 20D40 

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

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

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