Omar Antolín

For any prime p there are exactly two non-abelian groups of order p³:

1. The Heisenberg group of the field with the p elements, which consists of matrices of the form
[1 a c]
[0 1 b]
[0 0 1]
with a, b, c in Fₚ.

2. The semidirect product Z/p² ⋊ Z/p with Z/p acting on Z/p² via a·x := (1+ap)x.

Those two non-isomorphic groups have isomorphic character tables!

What are some other nice examples of infinite families of groups with isomorphic character tables?

#RepresentationTheory

Feb 17 at 11:07pmWeb
Show threadDima Pasechnik 🇺🇦 🇳🇱Feb 18
@oantolin terminology confusion? These groups are exponent 𝑝, and order 𝑝³
Show threadOmar AntolínFeb 18
@dimpase No, just a simple typo. I'll add the missing ³. Thanks for noticing the problem.
Show threadDima Pasechnik 🇺🇦 🇳🇱Feb 18
@oantolin there are several versions of this problem, depending on what exactly is meant by a character table - do you take powermaps into account (the knowledge, for each g in G, the column in the table for 𝑔ᵏ for each k.) ?
Show threadOmar AntolínFeb 18
@dimpase I had in mind just the minimum amount of information one would call the character table: the square matrix of values of the complex characters on the conjugacy classes of group elements; where two such matrices A and B are considered isomorphic if A = PBQ for some permutation matrices P and Q.
Show threadOmar AntolínFeb 18
@dimpase I learned from the paper that @antoinechambertloir linked (https://www.sciencedirect.com/science/article/pii/S0021869305004722), that groups with isomorphic tables+powermaps are called "Brauer pairs".
Show thread⏚ Antoine Chambert-LoirFeb 18
@oantolin https://www.sciencedirect.com/science/article/pii/S0021869305004722
Show threadOmar AntolínFeb 18
@antoinechambertloir Great, thank you!
Show threadDima Pasechnik 🇺🇦 🇳🇱Feb 18

@antoinechambertloir @oantolin
I recall discussing with LMFDB folks how to get short fingerprints for groups - and the conclusion was "well, tough luck, it would not really work"

https://www.lmfdb.org/Groups/Abstract/

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