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What we can say about two groups G and H when their group rings, $\mathbb{C}[G]$ and $\mathbb{C}[H]$, are Morita equivelent?

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I will assume that $G$ and $H$ are finite groups, since I don't know the theory otherwise. Then it just says that $G$ and $H$ have the same number of conjugacy classes.

Indeed, $\mathbb{C}[G]$ is Morita-equivalent to $\mathbb{C}^m$ where $m$ is the number of conjugacy classes of $G$, and $\mathbb{C}^m$ is Morita-equivalent to $\mathbb{C}^n$ iff $m=n$.

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    $\begingroup$ The question for infinite groups seems like a quite interesting question. I wonder if an answer is even known. $\endgroup$ – Tobias Kildetoft Apr 3 '16 at 14:22
  • $\begingroup$ Indeed, I have absolutely no clue. $\endgroup$ – Captain Lama Apr 3 '16 at 14:34
  • $\begingroup$ @CaptainLama tnx for ur answer, it's helpful. $\endgroup$ – Shakiba Apr 4 '16 at 12:15
  • $\begingroup$ @TobiasKildetoft tnx for being in discussion $\endgroup$ – Shakiba Apr 4 '16 at 12:24
  • $\begingroup$ @CaptainLama Do u have any reference in algebraic Morita equivalence? $\endgroup$ – Shakiba Apr 5 '16 at 7:04

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