[Paper Review] Hypergroups over the group and generalizations of Schreier's theorem on group extensions

Samuel H. Dalalyan|arXiv (Cornell University)|Mar 24, 2014

Advanced Topics in Algebra1 references3 citations

TL;DR

This paper introduces a novel concept of hypergroups over a group to generalize Schreier's theorem on group extensions. By defining a hypergroup structure on a right transversal to a subgroup, the authors establish a canonical bijection between isomorphism classes of group extensions of degree $ m $ and orbits of hypergroups under a group action, unifying and extending classical cohomological classification of extensions, particularly for non-abelian groups.

ABSTRACT

Let $H$ be a group, $m$ be a positive integer, $Ext_m H$ be the set of all isomorphic in $G$ classes of group monomorphisms $φ: H ightarrow G$ such that index of $φ(H)$ in $G$ is $m$. The main goal of this paper is to describe the elements of $Ext_m H$ 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 $Q$.

Motivation & Objective

• To generalize Schreier's 1926 theorem on group extensions to non-abelian groups using a new algebraic structure.
• To define and characterize hypergroups over a group as a unifying framework for groups, fields, and linear spaces.
• To establish a canonical bijection between isomorphism classes of group extensions of degree $ m $ and orbits of hypergroups under a group action.
• To extend the cohomological classification of extensions to non-abelian normal subgroups via two-dimensional cohomology.
• To recover Schreier's original theorem as a special case when the group $ H $ is abelian.

Proposed method

• Define a hypergroup over a group $ H $ using a right transversal $ M $ to a subgroup $ H $ in a group $ G $, with structural mappings $ heta = (\Phi, \Psi, \Xi, \Lambda) $.
• Introduce the notion of an (outer) exact product associated with a hypergroup, generalizing direct, semidirect, and Neumann's general products.
• Construct a group $ \mathcal{G} $ and an action $ \mathcal{A} $ on the set $ Hg(H,M) $ of isomorphism classes of hypergroups over $ H $.
• Define two-dimensional cochains $ \kappa \in H^M $, coboundaries $ \Lambda_\kappa $, and the cohomology group $ H^2((M,\Xi), H, \underline{\Psi}) $.
• Prove that $ B^2((M,\Xi), H, \underline{\Psi}) $ is a normal subgroup of $ Z^2((M,\Xi), H, \underline{\Psi}) $ when $ H $ is abelian, enabling the quotient cohomology group.
• Establish a canonical bijection $ Ext_m(H) \cong Hg(H,M)/\mathcal{A} $, and refine it to classify extensions by a given quotient group $ Q $.

Experimental results

Research questions

• RQ1 How can group extensions of a fixed degree $ m $ be classified using a unified algebraic structure?
• RQ2 Can Schreier's theorem on group extensions be generalized beyond abelian normal subgroups?
• RQ3 What is the role of hypergroups over a group in parametrizing group extensions?
• RQ4 How do structural mappings $ \Phi, \Psi, \Xi, \Lambda $ encode the group multiplication in the extension?
• RQ5 Under what conditions does the two-dimensional cohomology group $ H^2(Q, H, \underline{\Psi}) $ classify extensions of $ H $ by $ Q $?

Key findings

• There exists a canonical bijection between the set $ Ext_m(H) $ of isomorphism classes of group extensions of degree $ m $ and the quotient set $ Hg(H,M)/\mathcal{A} $ of hypergroups over $ H $ under the action $ \mathcal{A} $.
• The set $ Ext_{m,0}(H) $ of normal extensions is bijective to $ Hg_0(H,M)/\mathcal{A} $, where $ Hg_0 $ consists of hypergroups with trivial $ \Phi $-action.
• For a fixed isomorphism class $ [\Xi] $, the set $ Hg_0(H,M,[\Xi])/{\mathcal{A}} $ is bijective to $ Ext(H,Q) $, where $ Q = (M,\Xi) $.
• When $ H $ is abelian, the map $ j: Hg_0(H,M,\Xi,\Psi)/\mathcal{A}_0 \to H^2((M,\Xi), H, \underline{\Psi}) $ is a canonical bijection, recovering Schreier's theorem.
• The two-dimensional cohomology group $ H^2((M,\Xi), H, \underline{\Psi}) $ classifies extensions of $ H $ by $ Q $ when $ H $ is abelian, with the classification depending on the reverse homomorphism $ \underline{\Psi} $.
• The construction fails for non-abelian $ H $ in general due to non-normality of $ B^2 $ and non-commutativity, which prevents forming the quotient group.