Splitting of Clifford groups associated to finite abelian groups

TL;DR

This paper proves that the extension of Clifford groups associated with finite abelian groups splits as a semidirect product precisely when the group order is not divisible by four, confirming a conjecture and extending previous results.

Contribution

It establishes a necessary and sufficient condition for the splitting of Clifford group extensions for all finite abelian groups, generalizing prior cyclic group results.

Findings

Extension splits iff group order is not divisible by four

Confirms a conjecture by Korbelár and Tolar

Extends cyclic case to all finite abelian groups

Abstract

The Clifford group associated with a finite abelian group gives rise to a natural extension by the corresponding symplectic group. We prove that this extension splits as a semidirect product if and only if the group order is not divisible by four. This confirms a conjecture of Korbel\'{a}\v{r} and Tolar and extends their cyclic result to arbitrary finite abelian groups.