# Connection between $GL(\mathbb{Z}_p)$ and $GL(\mathbb{F}_p)$

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

-
Is $\Bbb Z_p$ the $p$-adic integers or something else? – Asaf Karagila Jan 12 '13 at 14:57

$\newcommand\Z{\mathbb{Z}}$ 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.

-
May I also suggest Hiller and Rhea's Automorphisms of Finite Abelian Groups for similar constructions. – Alexander Gruber Jan 12 '13 at 16:30
Alternatively google for "Cartan-Bauer triangle" or have a look at Serre's Linear Representation of Finite Groups, chapter 14-16. – BIS HD Nov 21 '13 at 11:34
@BISHD the Cartan-Brauer triangle concerns irreducible representations over $\mathbb{F}_p$, rather than indecomposable ones. That's a rather different question. – Alex B. Nov 21 '13 at 15:23