two questions in modular representation theory I have two questions: 


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*Let $G$ be a finite group. Because complex representations of $G$ are completely reducible to know all representations is same as knowing irreducible ones. In case of modules over group algebra ${\overline{\mathbb F_p}}[G]$ when $p|o(G),$ we know there are possibly indecomposable modules of arbitrary large degree. In such cases when do we say that we know "all" representations of $G$ with appropriate sense the word "all"?

*Is there any way or reference to compute order of group GL$_2(\mathbb Z/p^n\mathbb Z)$? I am trying to work with modules over $\overline{\mathbb F_p}[\mbox{GL}_2(\mathbb Z/p^n\mathbb Z)].$ Is it something very trivial or very difficult?


I request all to give answer in elementary language if possible.
 A: Mariano has addressed question (1), but let me add that finite representation type is extremely rare especially for interesting classes of groups like the simple nonabelian ones: in characteristic $p$, the Sylow $p$-subgroups must be cyclic.  Also, the initial work by Higman was refined by looking at individual $p$-blocks of a finite group and their representation type (Brauer, Dade, Janusz).   To get a block of tame representation type, you need $p=2$ while the block must have defect group of a very special type: dihedral, semidihedral, or generalized quaternion.    A general reference is:
MR1064107 (91c:20016) 20C20 (16G60 16G70)
Erdmann, Karin (4-OX),
Blocks of tame representation type and related algebras.
Lecture Notes inMathematics, 1428.
Springer-Verlag, Berlin, 1990. xvi+312 pp.
Concerning question (2), it really goes off in another direction and has been studied mainly by people interested in $p$-adic representation theory.   It would be useful for them to know more about representations of various linear groups over rings of $p$-adic integers or over finite residue rings other than the residue field such
as $\mathbb{Z}/p^n \mathbb{Z}$ when $n>1$.   This is a tough problem to attack using methods of finite group theory, whether you start by working over a ground field like $\mathbb{C}$ or else look at $p$-modular representations of these finite matrix groups.   As far as I know, results in this direction have been rather few and far between.
A: For (1), in a technical sense, for sufficiently complicated groups $G$ we never say we know all modular representations of $G$, because the classification problem is in that case wild.
A theorem of Higman tells us exactly when there are finitely many indecomposable modules in terms of the structure of Sylow subgroups: a necessary and sufficient condition is that the $p$-Sylow subgroups be cyclic—and if I recall correctly in that case one can in principle construct them all. 
There is a remaining case, that of groups of tame representation type, where there is a lot of technology available, and I would say that in that case we "know" the representation theory when you can make a picture of of Auslander-Reiten quiver of the group. 
