Hypergroups over the group and generalizations of Schreier's theorem on group extensions

Abstract

Let $H$ be a group, $m$ be a positive integer, $Ex{t}_{m}H$ be the set of all isomorphic in $G$ classes of group monomorphisms $\phi :H\to G$ such that index of $\phi (H)$ in $G$ is $m$. The main goal of this paper is to describe the elements of $Ex{t}_{m}H$ in terms of a new concept of hypergroups over the group. The obtained result is a very broad generalization of the Schreier theorem (1926). As an application, a series of intermediate generalizations is obtained; particularly, a description of the set $Ext(H,Q)$ of isomorphic classes of all extensions of a noncommutative group $H$ by $Q$.

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• Structures induced on transversals to a subgroup of a group and hypergroups over the group