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I'd like to ask the following question:

Let $G$ and $H$ be finite groups.

Is there a useful criterion involving the ordinary character table which assures that $G$ and $H$ are isomorphic as groups?

If the ordinary character tables are isomorphic, the groups $G$ and $H$ don't necessarily have to be isomorphic (e.g. $G=Q_8$ and $H=D_8$), but I wonder, if there is a criterion known, such like

"If the ordinary character tables of $G$ and $H$ are isomorphic and some other property holds, then $G$ and $H$ are isomorphic".

Thanks for the help.

Bernhard Boehmler
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  • For example, $Q_8$ and $D_8$ can be distinguished by their determinant representations, but I don't know a general criterion. – Bernhard Boehmler Mar 15 '20 at 17:22
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    These questions are related: https://mathoverflow.net/questions/11306/character-table-does-not-determine-group-vs-tannaka-duality https://mathoverflow.net/questions/500/finite-groups-with-the-same-character-table . – LSpice Mar 15 '20 at 18:49
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    In particular, https://mathoverflow.net/a/11311 (and other higher-tech versions of it under the same question) seem to be ways of answering your question, though maybe not in the spirit you want. – LSpice Mar 15 '20 at 18:50
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    There is a large literature on this subject. One person who has strong results around this question is A. Turull. In general, the character table can be quite far from determining the isomorphism type of the group. Another paper on this topic is a per of E.C. Dade in the very first issue of Journal of Algebra (1964), which gave a negative answer to a question of R. Brauer about character tables and power maps. – Geoff Robinson Mar 16 '20 at 14:03
  • @GeoffRobinson Thank you very much for the comment. I wrote an e-mail to A. Turull and he informed me that his former student A. Nenciu did research in the same direction. In her PhD thesis (see http://etd.fcla.edu/UF/UFE0014824/nenciu_a.pdf) in Proposition 3.0.14 there is another criterion. – Bernhard Boehmler Mar 22 '20 at 14:57
  • Thanks. After you asked the question, I remember it was probably one of Alex's students who worked on this. – Geoff Robinson Mar 22 '20 at 15:47

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