  The distinguishing number of a graph was introduced by Albertson and Collins
as a measure of the amount of symmetry contained in the graph. Tymoczko
extended this definition to faithful group actions on sets; taking the set to
be the vertex set of a graph and the group to be the automorphism group of the
graph allows one to recover the previous definition. Since then, several
authors have studied properties of the distinguishing number as well as
extensions of the notion. In this paper, we first answer a few open questions
regarding the distinguishing number. Next we turn to generalizations regarding
the labeling of Cartesian powers of a set and the different subgroups that can
be obtained through labelings. We then introduce a new partially ordered set on
partitions that follows naturally from extending the theory of distinguishing
numbers to that of distinguishing partitions. Then we investigate the groups
obtainable from partitioning Cartesian powers of a set in more detail and show
how the original notion of the distinguishing number of a graph can be
recovered in this way. Next, we introduce a polynomial and a symmetric function
generalization of the distinguishing number. Finally, we present a large number
of open questions and problems for further research.
