TITLE: Connection between $GL(\mathbb{Z}_p)$ and $GL(\mathbb{F}_p)$
QUESTION [5 upvotes]: Suppose there is a finite group $G$. Is there a connection between indecomposable representations over $\mathbb{Z}_p$ and $\mathbb{Z}/p \mathbb{Z}$. I know what to do if $G$ is cyclic. But if not?
If this is not possible: Suppose $G$ has a normal subgroup. So induction would be possible. Is there literature (which is not too difficult)
for such an induction, where the representations are not over a field (like in Clifford's theorem), but over a ring?
Best redards

REPLY [4 votes]: $\newcommand\Z{\mathbb{Z}}$
There is a connection between indecomposable representations over $\Z_p$ and over $\Z/p^n\Z$ for all $n$, which goes as follows: clearly, if $M$ is a $\Z_p[G]$-module such that $M/p^nM$ is an indecomposable $(\Z/p^n\Z)[G]$-module for some $n$, then $M$ is indecomposable (since any decomposition of $M$ would descend to a decomposition of any reduction). The converse is a theorem:
Theorem: A $\Z_p[G]$-module $M$ is indecomposable if and only if $M/p^nM$ is indecomposable for some $n$.
This $n$ might have to be taken greater than 1 to really get "if and only if". Also
Theorem: If $M,N$ are two $\Z_p[G]$-modules, then $M\cong N$ if and only if $M/p^kM$ and $N/p^kN$ are isomorphic $(\Z/p^k\Z)[G]$-modules for some explicitly computable $k$.
These results and more can be found in Curtis and Reiner, Representation theory of finite groups and associative algebra, §76.