Finite 2-groups with odd number of conjugacy classes

Jaikin-Zapirain, Andrei; Tent, Joan

doi:10.1090/tran/7067

Finite 2-groups with odd number of conjugacy classes
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by Andrei Jaikin-Zapirain and Joan Tent PDF

Trans. Amer. Math. Soc. 370 (2018), 3663-3688 Request permission

Abstract:

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite $2$-groups with exactly $k$ real conjugacy classes. On the other hand we construct infinitely many finite $2$-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-$p$ techniques, and, in particular, we use the Kneser classification of semi-simple $p$-adic algebraic groups.

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