Enumeration of lifts of commuting elements of a group

TL;DR

This paper provides a formula for counting the lifts of commuting elements in a group through a group epimorphism with finite kernel, and applies it to specific cases including the quaternion group.

Contribution

It introduces a new formula for enumerating lifts of commuting elements and applies it to groups with quaternion kernels, revealing specific possible counts.

Findings

Number of lifts is finite and takes values 0, 8, 16, 24, or 40 for certain kernels.

All these lift counts are realizable by some groups and epimorphisms.

The formula relates the count to representations of the kernel.

Abstract

Given commuting elements a, b of a group G and a group epimorphism q : G' \to G with finite kernel, the set of commuting lifts of a, b to G' is finite (possibly, empty). The second named author obtained a formula for the number of such lifts in terms of representations of Ker q. We apply this formula to several group epimorphisms q with the same kernel. In particular, we analyze the case where Ker q = Q_8 is the quaternion group. We show that in this case the number in question is equal to 0, 8, 16, 24, or 40. We show that all these numbers are realized by some G,G',q,a,b.