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Jan 26, 2021 at 8:20 comment added YCor Maybe a more definite negative answer (for fixed $n\ge 2$) would be to show that the characterization does not only depend on the conjugacy classes of $A$, $B$, and $C$.
Jan 26, 2021 at 8:08 history edited Federico Poloni CC BY-SA 4.0
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Jan 26, 2021 at 8:01 history answered Federico Poloni CC BY-SA 4.0