  In this paper, we consider a class of symmetry groups associated to
communication channels, which can informally be viewed as the transformations
of the set of inputs that ``commute'' with the action of the channel. These
groups were first studied by Polyanskiy in (IEEEToIT 2013). We show the simple
result that the input distribution that attains the maximum mutual information
for a given channel is a ``fixed point'' of its group. We conjecture (and give
empirical evidence) that the channel group of the deletion channel is extremely
small (it contains a number of elements constant in the blocklength). We prove
a special case of this conjecture. This serves as some formal justification for
why the analysis of the binary deletion channel has proved much more difficult
than its memoryless counterparts.
