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