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As a consequence of THEOREM 2.6 in [ROBERT J. BLATTNER, SUSAN MONTGOMERY, CROSSED PRODUCTS AND GALOIS EXTENSIONS OF HOPF ALGEBRAS, PACIFIC JOURNAL OF MATHEMATICS Vol. 137, No. 1, 1989, 37-54], we know that:

Let $p$ be a prime, $H$ be a normal subgroup of $G$ with $(p,|G:H|)=1$ and $V$ be a left $kG$-module, where the characteristic of $k$ is $p$. If $V$ is a semisimple $kH$-module, then $V$ is a semisimple $kG$-module.

This consequence tells us that sometimes the complete reducibility can be deduced from the restricted representation.

Question: Besides the above consequence, are there any other similar theorems?

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It is probably obvious to you, but I think it is should be mentioned that the other way round, you have Clifford's theorem in modular representation theory of finite groups : Theorem (Clifford) If $H$ is any normal subgroup of $G$ and $V$ is a semi-simple $k[G]$-module, then $V_{|H}$ is a semi-simple $k[H]$-module. –  Niels Jun 5 '11 at 8:04
    
Yes. But my question is that: if the restricted representation $V|_{H}$ is semisimple, in what conditions can we deduce that $V$ is semisimple? Here $H$ need not be a normal subgroup. Furthermore, we consider several restricted representations $\{V|_{H_{i}}\}$. If the restricted representations $\{V|_{H_{i}}\}$ are semisimple, in what conditions can we deduce that $V$ is semisimple? –  sife Jun 5 '11 at 10:45
    
@sife: Your last question needs a more precise formulation concerning the indices of the subgroups relative to the given prime number. Like the original question asked, this further question seems too loose to answer directly. –  Jim Humphreys Jun 5 '11 at 11:36

1 Answer 1

The statement quoted here is not due originally to Blattner and Montgomery, but is a familiar generalization of the standard "averaging" proof of Maschke's Theorem on complete reducibility for finite groups: see for example Exercise 8 in Section 10 of the book Representations of Finite Groups and Associative Algebras by Curtis and Reiner (1962). As they note, the group $G$ could be infinite.

It's also worth noting Section 62 in that book, where it is shown that the group algebra of a finite group is a Frobenius algebra (indeed, group algebras illustrate the general fact that a finite dimensional Hopf algebra is always Frobenius). Using the language of homological algebra, one sees then how to generalize Maschke's theorem and other results about finite groups to Frobenius algebras. This becomes an important theme in the work of Blattner-Montgomery and others on Hopf algebras.

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