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Each complex representation of a finite group is semisimple (i.e. decomposes into a direct sum of irreducible complex representations).

Question: What are the countable groups satisfying the same property?

Remark: We do not restrict to finite dimensional representations.

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The characterization is two-way: all $\mathbb C$ representations are semisimple iff $G$ is finite.
(Corollary page 660 in Connell, Ian G. On the group ring. Canad. J. Math. 15 1963 650--685)

This is essentially what Maschke's theorem is about. I don't think finite dimensionality of representations comes into play.

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  • $\begingroup$ Or something might be escaping my notice... this is what the question seems like to me. You might have to tell me why it isn't this simple. $\endgroup$ – rschwieb Apr 11 '18 at 11:11
  • $\begingroup$ I don't see an 'iff' in Maschke's theorem. How does the converse follow from it? $\endgroup$ – lisyarus Apr 11 '18 at 11:14
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    $\begingroup$ @lisyarus Hmm, interesting, I didn't realize the wiki article only apparently has "the classical statement." The two-way statement appears in Connell's On the group ring and Lam's First course in noncommutative rings Theorem 6.1 $\endgroup$ – rschwieb Apr 11 '18 at 11:20
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    $\begingroup$ @lisyarus: The generalized Maschke theorem is here Corollary page 660. $\endgroup$ – Sebastien Palcoux Apr 11 '18 at 12:58
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    $\begingroup$ @rschwieb Thank you a lot! $\endgroup$ – lisyarus Apr 11 '18 at 13:18

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