  The fixing number of a graph $G$ is the smallest cardinality of a set of
vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex
in $S$. The fixing set of a group $\Gamma$ is the set of all fixing numbers of
finite graphs with automorphism group $\Gamma$. Several authors have studied
the distinguishing number of a graph, the smallest number of labels needed to
label $G$ so that the automorphism group of the labeled graph is trivial. The
fixing number can be thought of as a variation of the distinguishing number in
which every label may be used only once, and not every vertex need be labeled.
We characterize the fixing sets of finite abelian groups, and investigate the
fixing sets of symmetric groups.
