Base Size Sets and Determining Sets

TL;DR

This paper investigates the base size and determining sets of various finite groups, providing explicit characterizations for abelian and dihedral groups, and highlighting differences between these sets in certain cases.

Contribution

It offers new characterizations of base size and determining sets for specific group families, including finite abelian and dihedral groups, and identifies cases where these sets differ.

Findings

For finite abelian groups, B(G)=D(G)={1,2,...,k}

Explicit descriptions of B(G) and D(G) for dihedral groups D_{p^k} and D_{2p^k}

B(G) is not equal to D(G) for dihedral groups D_{pq} with distinct odd primes p and q

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.