Enumeration of lifts of commuting elements of a group

Details Enumeration of lifts of commuting elements of a group Enumeration of lifts of commuting elements of a group

2009-04-24

arxiv.org/abs/0904.3902v1

Mathematics

Group Theory

14 pages

Scientific paper

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.

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