Difference sets disjoint from a subgroup

2017/03/20 by Hoagland, Courtney, Humphries, Stephen P., Poulsen, Seth
#05B10 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1703.06979

Abstract

We study finite groups G having a subgroup H and D ⊂ G ∖ H such that the multiset \ xy^-1:x,y ∈ D\ has every non-identity element occur the same number of times (such a D is called a \it difference set). We show that H has to be normal, that |G|=|H|², and that |D ∩ Hg|=|H|/2 for all g ∉ H. We show that H is contained in every normal subgroup of prime index, and other properties. We give a 2-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets.

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