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Transactions of the American Mathematical Society

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Finite 2-groups with odd number of conjugacy classes


Authors: Andrei Jaikin-Zapirain and Joan Tent
Journal: Trans. Amer. Math. Soc. 370 (2018), 3663-3688
MSC (2010): Primary 20D15; Secondary 20C15, 20E45, 20E18
DOI: https://doi.org/10.1090/tran/7067
Published electronically: December 27, 2017
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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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Andrei Jaikin-Zapirain
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, and Instituto de Ciencias Matemáticas - CSIC, UAM, UCM, UC3M, 28049 Madrid, Spain
Email: andrei.jaikin@uam.es

Joan Tent
Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
Email: joan.tent@uv.es

DOI: https://doi.org/10.1090/tran/7067
Keywords: 2-groups, real classes, $p$-adic groups
Received by editor(s): October 1, 2015
Received by editor(s) in revised form: August 19, 2016
Published electronically: December 27, 2017
Additional Notes: This paper was partially supported by the grant MTM 2011-28229-C02-01 and MTM2014-53810-C2-01 of the Spanish MEyC and by the ICMAT Severo Ochoa project SEV-2011-0087
The second author was supported by PROMETEOII/2015/011
Article copyright: © Copyright 2017 American Mathematical Society

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