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What are the necessary and sufficient conditions for two finite groups $G$ and $H$ to have same complex-valued character table? Is there any criterion for which one could know about the character table similarity of two finite groups without direct computations of each table?

Obviously two isomorphic groups have same character table, but according to the case of $Q_8$ and $D_8$, I'm searching for a weaker criterion.

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s/to/two/ # That is what he said... –  jmc May 18 '13 at 18:13
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2 Answers

up vote 6 down vote accepted

Finite groups have the same complex character tables if and only if their group algebras are isomorphic as quasi-Hopf algebras (if and only if the group algebras are twisted forms of each other as Drinfel'd quasi-bialgebras, if and only if there is non-associative bi-Galois algebra over these groups). For details see here: rxiv.org/abs/math/0001119.

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This is a complicated question. A pair of non-isomorphic groups with the same character table is sometimes called a "Brauer Pair". There are many such pairs, especially among $p$-groups.

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Thanks Geoff, but what I mean is : for two given groups G and H, without computing the character tables, how one could determine G and H are "Brauer Pair" or not? Is there any criterion for that? –  Farrokh Shirjian May 18 '13 at 19:15
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The usual definition of "Brauer pair" is more technical. Given a group G and a positive integer n, there is a map from the classes of G to itself, where the class of x goes to the class of x^n. These are the "power maps". Brauer asked in his famous collection of problems whether two groups with identical character tables and power maps must be isomorphic. The answer turns out to be "no". The first counterexample was given by Dade. Pairs of nonisomorphic groups with identical character tables and power maps are now called "Brauer pairs". Usually, the phrase is applied to p-groups. –  Marty Isaacs Nov 4 '13 at 21:09
    
@MartyIsaacs Thank you so much for your useful description Marty. –  Farrokh Shirjian Nov 10 '13 at 22:24
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