In this talk, we will present some results that link the
structure of a universal algebra with its clone of
polynomial functions. It has recently been
proved that for every finite algebra
with a Mal'cev term, the clone of polynomial operations
and the clone of term operations are both finitely related.
This establishes that up to isomorphism
and term equivalence, there are only countably many finite
algebras with a Mal'cev term; in
group theory this yields that for every group G, there exists
a subgroup H of some finite power G^k such that for
all n, all subgroups of G^n can - in a certain way -
be constructed from H.
We will compare two concepts of nilpotence for
expansions of groups. Starting from the well-known fact
that every finite nilpotent group is a direct product
of p-groups and Kearnes's generalization to
finite nilpotent algebras in congruence modular varieties,
we present a decomposition result for certain nilpotent
infinite expanded groups.