$\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$ I'm trying to read (part of) "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)).
Here are some basic assertions I need help with:
(by context $p$ is a prime number and $G$ is a finite group)

For each $p$ dividing $|G|$, and each irreducible $\mathbb{F}_p G$-module $M$, let $E=\text{Hom}_G(M,M)$. Then:
  1. $\text{Hom}_G(\mathbb{F}_p G,M) \cong E^{r_M}$,
  2. $H^1(G,M) \cong E^{s_M}$,
  for certain non-negative integers $r_M$, $s_M$.

I am familiar with all terms except for "irreducible module". I know what a simple module is, and what an irreducible representation is. My guess is that irreducible module is just an old term for a simple module, but I'm not sure. So, I need help with this terminology issue and with the proof of both assertions.
Also note that this is my first encounter with group cohomlogy. I read the definition here: http://groupprops.subwiki.org/wiki/First_cohomology_group. I used to know a bit about cohomology in a topological context, but it's not fresh in my mind.
 A: First of all, if you are not familiar with group cohomology, $H^1(G,M)$ means $\operatorname{Ext}^1_{\mathbb{F}_pG}(\mathbb{F}_p,M)$ and irreducible means simple.
By Schur's lemma $E$ is a division ring over $\mathbb{F}_p$, in fact a finite field.  Now note that for any $\mathbb{F}_p$-module $N$, both $\hom_{\mathbb{F}_p G}(N,M)$ and $\operatorname{Ext}_{\mathbb{F}_pG} (N,M)$ are $E$-modules by functoriality.  Since any $E$-module is free, both sorts of groups are isomorphic (as abelian groups, or $E$-modules, or $\mathbb{F}_p$-modules) to a direct sum of $E$s.
Edit: $\hom_{\mathbb{F}_pG}(N,-)$ and $\operatorname{Ext}^1_{\mathbb{F}_pG}(N,-)$ are functors from the category of $\mathbb{F}_pG$-modules to the category of $\mathbb{F}_p$ vector spaces.  How you describe functoriality of Ext depends on the way you choose to construct it, but it is a derived functor so it is a functor which ever way you do it :)
For example, you may construct Ext in such a way that elements of $\operatorname{Ext}^n_{\mathbb{F}_pG}(N,M)$ are equivalences classes of module maps $\phi: P_n \to M$, where $P_n$ is a certain projective resolution of $N$.  Then if $f:M\to M$ is a module map, the class of $\phi$ acted on by $f$ is represented by $f \circ \phi$.
Alternatively, you may think of $\operatorname{Ext}^n_{\mathbb{F}_pG}(N,M)$ as consisting of equivalence classes of long exact sequences 
$$ 0 \to M \to L_n \to \cdots \to L_1 \to N \to 0 $$
of $\mathbb{F}_pG$-modules beginning with $M$ and ending with $N$.  The action of $f$ on the equivalence class of such a sequence is represented by a sequence as follows
$$ \begin{array}{cccccc}
0 & \to & M & \to L_n & \to L_{n-1} & \to \cdots \\
  & & \downarrow f &\downarrow & & \\
0 & \to & M & \to X & \to L_{n-1} & \to \cdots
\end{array}
$$
where $X$ is the pushout of $M \to L_n$ and $f:M \to M$.
