  Symmetry arguments are frequently used -- often implicitly -- in mathematical
modeling of natural selection. Symmetry simplifies the analysis of models and
reduces the number of distinct population states to be considered. Here, I
introduce a formal definition of symmetry in mathematical models of natural
selection. This definition applies to a broad class of models that satisfy a
minimal set of assumptions, using a framework developed in previous works. In
this framework, population structure is represented by a set of sites at which
alleles can live, and transitions occur via replacement of some alleles by
copies of others. A symmetry is defined as a permutation of sites that
preserves probabilities of replacement and mutation. The symmetries of a given
selection process form a group, which acts on population states in a way that
preserves the Markov chain representing selection. Applying classical results
on group actions, I formally characterize the use of symmetry to reduce the
states of this Markov chain, and obtain bounds on the number of states in the
reduced chain.
