  This paper generalizes the basic notions of additive and multiplicative
combinatorics to the setting of group actions: if $G$ is a group acting on a
set $X$, and we have subsets $A\subseteq G$ and $Y\subseteq X$ such that the
set of pairs $g\cdot y$ with $g\in A,y\in Y$ is not much larger than $Y$, what
structure must $A$ and $Y$ have?
  Briefly, what is the structure of sets with small "image set"?
  In this setting, we develop analogs of Ruzsa's triangle inequality, covering
theorems, multiplicative energy, and the Balog-Szemer\'{e}di-Gowers theorem.
Approximate stabilizers, which we call symmetry sets, play an important role.
  While our focus is on presenting a general theory, we answer the inverse
image set question in some special cases.
  To do so, we combine the group action version of the
Balog-Szemer\'{e}di-Gowers theorem with structure theorems for approximate
groups and bounds for the sizes of symmetry sets.
