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Let $G$ be a finite group and $H$ a subgroup whose index is prime to $p$. Suppose $V$ is a finite-dimensional representation of $G$ over $\mathbb{F}_p$ whose restriction to $H$ is semisimple. Prove that $V$ is semisimple.

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    "prime to $p$" means "not divisible by $p$"?2012-05-29
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    @rschwieb: It's short for saying the index and $p$ are coprime, which for prime $p$ is equivalent to not-divisible-by-$p$.2012-05-29
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    @anon I gathered that much, I just have never heard "prime to $p$" used in place of "coprime with $p$"2012-05-29
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    I can deduce a few things, but I confess I don't know much about modular forms. Is there any special reason that if $p$ divides the order of $H$, that a semisimple $H$ representation would also be a semisimple $G$ representation?2012-05-30
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    If I remember correctly, you can use the usual averaging argument for proving Maschke's theorem (just take the average over coset representatives of $H$ instead of over all elements of $G$)2012-05-30
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    @Jug: Yes you can do it that way if you are careful!2012-05-30
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    Given that this implies Maschke's theorem jug's idea is probably the way to go.2012-06-08
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    (12.8) in Aschbacher's book "Finite Group Theory" (directly before Maschke's theorem) proves the statement for $H$ a $p$-Sylow (considering also (12.6)). If someone feels like, please expand it into an answer...2012-06-08

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(Expanded form of the answers from the comments.)

A module is semisimple iff every submodule has a direct complement.

Aschbacher's Finite Group Theory on pages 39-40 proves that if U is a sub FG-module of V and U has an FP-module direct complement, then it has an FG-module direct complement where P is a Sylow p-subgroup of G. The argument is an averaging argument as described.

Since H has index coprime to p, H contains a Sylow p-subgroup P of G. Every semisimple FH-module is also semisimple as an FP-module (or just replace n in Aschbacher's proof with $[G:P]$ and allow $P=H$ to be any subgroup of index coprime to p), and so Aschbacher's result answers your question.


The same proof is phrased in more complicated language on pages 70, 71, and 72 of Benson's Representations and Cohomology Part 1.

A module is called relatively H-projective if G-module homomorphisms that split as H-module homomorphisms also split as G-module homomorphisms. In other words, you just want to show that every G-module is relatively H-projective when $[G:H]$ is invertible in the ring, which is Corollary 3.6.9.

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    This is given as an exercise in Isaacs's CT and Curtis–Reiner, and when H=1 it is proven in many texts (Isaacs, Curtis–Reiner, Gorenstein, Huppert, Weintraub, Hilton).2012-06-13
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    Alperin's Local Rep Theory and the two-volume Curtis–Reiner book also give the relative projective proof. @user32134, let me know if you want google links to those proofs. Alperin's is pretty easy to read (page 66). The two-volume C-R is more like Benson's.2012-06-13