  In this paper, we investigate the stabilizers of certain multisets $\mod p^k$
with respect to their natural multiplicative action, completely describing them
for a certain family of polynomials whenever $p$ is an odd prime. This
elucidates an underlying structure on the level of elements that yields a new
proof of the classical Kummer Congruences. Hence the Kummer Congruences can be
viewed as a consequence of putting together this local information, the
interactions between which are understood. This gives a new perspective in that
all known approaches are based directly on global considerations, namely either
through global sum identities or $p$-adic integrals.
