Splitting of Clifford groups associated to finite abelian groups

Abstract

The Clifford group associated with a finite abelian group fits into an extension of the symplectic group by the underlying phase space. We prove that this extension splits as a semidirect product if and only if the order of the underlying abelian group is not divisible by four. The obstruction to splitting is controlled entirely by the 2-primary component and vanishes precisely when this component is trivial or cyclic of order two. This confirms a conjecture of Korbel\'a\v{r} and Tolar, extending their cyclic result to arbitrary finite abelian groups.