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I have a (probably very straightforward) question about blocks and simple modules. The problem I'm having is on p103 of Local representation theory by JL Alperin.

Let $G$ be a finite group. Let $B$ be a block of $G$ with defect group $D$. Let $b$ be the block of $N_G(D)$ which is the Brauer correspondent of $B$. Let $S$ be a simple $kN_G(D)$-module lying in $b$.

Apparently, $S$ must be a $k[N_G(D)/D]$-module, but I don't see this. Why is this true?

Thanks.

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1 Answer 1

up vote 3 down vote accepted

Restrict $S$ to $D$. What you get is semisimple by Clifford ($D$ is normal in $N_G(D)$). Now, $D$ is a $p$-group, hence the only simple module is trivial. Thus $D$ acts trivially on $S$, which is all you need.

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Could you very quickly explain why $S_D$ is a semisimple $kD$-module? -- Never mind, I see this now -- this is a simple application of Clifford's theorem. –  Clinton Boys Sep 11 '11 at 10:53
    
I edited....... –  mt_ Sep 11 '11 at 10:56

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