# Blocks and covering modules

I have a question relating to p109 of Local representation theory by JL Alperin.

Let $G$ be a finite group and let $N$ be a normal subgroup. If $B$ is a block of $G$, why must $B$ be a summand of $(k_{N\times N})^{G\times G}$?

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I think you hae either forgotten some hypotheses, or written the wrong question down. The statement as it stands is false if $N = G$ and $p$ divides $|G|$ ($p$ = char($k$)). – Geoff Robinson Sep 12 '11 at 5:00
I want to apply Exercise 9.6 and Lemma 9.7 to $B$. If I can show that $B$ has trivial source (as a $k(G\times G)$-module), then an indecomposable summand of $B_{N\times N}$ must have a vertex containing the intersection of $N\times N$ with a vertex of $B$, by Lemma 9.7. Exercise 9.6 says this happens, i.e. $B$ has trivial source, if $B\mid (k_{N\times N})^{G\times G}$. So it's this that I'm trying to prove. – Clinton Boys Sep 12 '11 at 7:51
@Clinton: I don't see how to interpret that line of the proof other than how you did, but how you did is definitely wrong. You should rephrase this question (and answer it if you know the answer!) so that it more specifically addresses your question: how can you use Lemma 9.7 and exercise 9.6 in this situation (since the hypotheses are not satisfied). – Jack Schmidt Sep 25 '11 at 16:23