  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.
