Base Size Sets and Determining Sets

2014/07/22 by Joshua D. Laison, Laison, Joshua D., Erin M. McNicholas +3
Mathematics · #05C25 #20B05 #20B25 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:05C25 #msc:20B05 #msc:20B25

paper · pdf · doi:10.48550/arxiv.1407.6046

10 pages, 1 figure

arxiv created 2014/07/22 · arxiv updated 2014/07/24

Abstract

Bridging the work of Cameron, Harary, and others, we examine the base size set B(G) and determining set D(G) of several families of groups. The base size set is the set of base sizes of all faithful actions of the group G on finite sets. The determining set is the subset of B(G) obtained by restricting the actions of G to automorphism groups of finite graphs. We show that for finite abelian groups, B(G)=D(G)=1,2,...,k where k is the number of elementary divisors of G. We then characterize B(G) and D(G) for dihedral groups of the form D_p^k and D_2p^k. Finally, we prove B(G) is not equal to D(G) for dihedral groups of the form D_pq where p and q are distinct odd primes.

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