  We propose a new interpretation of measures of information and disorder by
connecting these concepts to group theory in a new way. Entropy and group
theory are connected here by their common relation to sets of permutations. A
combinatorial measure of information and disorder is proposed, in terms of
integers and discrete functions, that we call the integer entropy. The Shannon
measure of information is the limiting case of a richer, more general
conceptual structure that reveals relations among finite groups, information,
and symmetries. It is shown that the integer entropy converges uniformly to the
Shannon entropy when the group includes all permutations, the Symmetric group,
and the number of objects increases without bound. The harmonic numbers have a
well-known combinatorial meaning as the expected number of disjoint, non-empty
cycles in permutations of n objects, and since integer entropy is defined in
terms of the expected value of the number of cycles over the set of
permutations, it also has a clear combinatorial meaning. Since all finite
groups are isomorphic to subgroups of the Symmetric group, every finite group
has a corresponding information functional, analogous to the Shannon entropy
and a number series analogous to the harmonic numbers. The Cameron-Semeraro
cycle polynomial is used to analyze the integer entropy for finite groups, and
to characterize the series analogous to the Harmonic numbers. We introduce and
use a reciprocal polynomial, the transposition polynomial that provides an
additional tool and new insights. Broken symmetries and conserved quantities
are linked through the cycle and transposition properties of the groups, and
can be used to generalize the analysis of stochastic processes.
