Extending structures I: the level of groups

Details Extending structures I: the level of groups Extending structures I: the level of groups

2010-11-07

arxiv.org/abs/1011.1633v2

Mathematics

Group Theory

24 pages

Scientific paper

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 $\cdot$ that can be defined on $E$ such that $H$ is a subgroup of $(E, \cdot)$. Thus we solve at the level of groups what we have called the \emph{extending structures problem}. A general product, which we call the unified product, is constructed such that both the crossed product and the Takeuchi's bicrossed product of two groups are special cases of it. It is associated to $H$ and to a system $\bigl((S, 1_S, \ast), \triangleleft, \, \triangleright, \, f \bigl)$ called a group extending structure and we denote it by $H \ltimes S$. There exists a group structure $\cdot$ 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 $\bigl((S, 1_S, \ast), \triangleleft, \, \triangleright, \, f \bigl)$. All such group structures $\cdot$ 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 general Schreier theorem is proved and an answer to a question of Kuperberg is given, both being special cases of our classification result. The above construction is related to the existence of hidden symmetries of $H$-principal bundles at the level of 0-dimensional manifolds (discrete sets).

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