TITLE: p-adic representations of groups
QUESTION [5 upvotes]: Hi, 
is there a possibility to classify irreducible representations of (finite) groups over $ \mathbb{Z_p} $
with $p\neq 2$ a prime? Can one get the number of the irred. representations  here? I tried to solve this for "easy" groups like the cyclic or symmetric groups, but did not really get a result. Maybe there is a connection to representation over $\mathbb{Z}/p\mathbb{Z}$?
Best regards

REPLY [6 votes]: What Alex says is correct, but there is another viewpoint, which is a bit easier in some ways.
You can consider finitely generated $\mathbb{Z}_{p}$-free $\mathbb{Z}_{p}G$-modules which have no proper non-zero pure submodules. These are the modules which are irreducible when the ground ring is extended to $\mathbb{Q}_{p}.$ There are finitely many isomorphism types of such modules. It may be possible to deduce the structure of faithful such modules for the dihedral group using Clifford theory.