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I'm tasked with proving that if I have a finite group G with a unique minimal normal subgroup, and a field F with characteristic not dividing the order of G, then there exists a faithful irreducible F-representation.

I can be fairly certain that the proof is to use Maschke's Theorem, since all the elements are there, but that normal subgroup also makes me think of Clifford's Theorem. My first thought was to take the regular FG-module and decompose with Maschke's, but I'm not sure what to do with that. I would be very grateful for any hints you can provide.

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up vote 5 down vote accepted

Suppose not. Then all the irreducible reps contain the unique minimal normal subgroup in their kernel. What does this tell you about the regular representation?

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Oh of course, because the kernel of a homomorphism is normal, and so a non-trivial kernel must contain our minimal normal subgroup. And this would then suggest that the regular FG-module is not faithful, which isn't true. Thank you! – Murray Jun 19 '12 at 10:36
Make sure to use Maschke to conclude the regular rep is a direct sum of irreducibles, so not faithful. The regular rep of a cyclic group of order 4 over a field of order 2 is faithful (and the group has a unique minimal normal subgroup), but it has no faithful irreducible representations. – Jack Schmidt Jun 19 '12 at 13:37

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