  In this paper we formalize the notions of information elements and
information lattices, first proposed by Shannon. Exploiting this formalization,
we identify a comprehensive parallelism between information lattices and
subgroup lattices. Qualitatively, we demonstrate isomorphisms between
information lattices and subgroup lattices. Quantitatively, we establish a
decisive approximation relation between the entropy structures of information
lattices and the log-index structures of the corresponding subgroup lattices.
This approximation extends the approximation for joint entropies carried out
previously by Chan and Yeung. As a consequence of our approximation result, we
show that any continuous law holds in general for the entropies of information
elements if and only if the same law holds in general for the log-indices of
subgroups. As an application, by constructing subgroup counterexamples we find
surprisingly that common information, unlike joint information, obeys neither
the submodularity nor the supermodularity law. We emphasize that the notion of
information elements is conceptually significant--formalizing it helps to
reveal the deep connection between information theory and group theory. The
parallelism established in this paper admits an appealing group-action
explanation and provides useful insights into the intrinsic structure among
information elements from a group-theoretic perspective.
