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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.

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  • $\begingroup$ For example, $Q_8$ and $D_8$ can be distinguished by their determinant representations, but I don't know a general criterion. $\endgroup$
    – Bernhard Boehmler
    Commented Mar 15, 2020 at 17:22
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    $\begingroup$ These questions are related: mathoverflow.net/questions/11306/… mathoverflow.net/questions/500/… . $\endgroup$
    – LSpice
    Commented Mar 15, 2020 at 18:49
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    $\begingroup$ In particular, 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. $\endgroup$
    – LSpice
    Commented Mar 15, 2020 at 18:50
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    $\begingroup$ 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. $\endgroup$
    – Geoff Robinson
    Commented Mar 16, 2020 at 14:03
  • $\begingroup$ @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 etd.fcla.edu/UF/UFE0014824/nenciu_a.pdf) in Proposition 3.0.14 there is another criterion. $\endgroup$
    – Bernhard Boehmler
    Commented Mar 22, 2020 at 14:57

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