Profinite groups with few conjugacy classes of 𝑝-elements

Wilson, John

doi:10.1090/proc/15925

Profinite groups with few conjugacy classes of $p$-elements
HTML articles powered by AMS MathViewer

by John S. Wilson

Proc. Amer. Math. Soc. 150 (2022), 3297-3305

DOI: https://doi.org/10.1090/proc/15925

Published electronically: April 1, 2022

PDF | Request permission

Abstract:

It is proved that a profinite group $G$ has fewer than $2^{\aleph _0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow $p$-subgroups are finite. (Here, by a $p$-element one understands an element that either has $p$-power order or topologically generates a group isomorphic to $\mathbb {Z}_p$.) A weaker result is proved for $p=2$.

References

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie, Berliner Sitzungsber. (1903), 987–991.
Daniel Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co., New York, 1980. MR 569209
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592, DOI 10.1090/surv/040.1
Wolfgang Herfort, An arithmetic property of profinite groups, Manuscripta Math. 37 (1982), no. 1, 11–17. MR 649560, DOI 10.1007/BF01239941
Andrei Jaikin-Zapirain and Nikolay Nikolov, An infinite compact Hausdorff group has uncountably many conjugacy classes, Proc. Amer. Math. Soc. 147 (2019), no. 9, 4083–4089. MR 3993799, DOI 10.1090/proc/14507
Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. I. Finite groups, J. Algebra 105 (1987), no. 2, 484–505. MR 873681, DOI 10.1016/0021-8693(87)90211-0
Aner Shalev, On almost fixed point free automorphisms, J. Algebra 157 (1993), no. 1, 271–282. MR 1219668, DOI 10.1006/jabr.1993.1100
John S. Wilson, On the structure of compact torsion groups, Monatsh. Math. 96 (1983), no. 1, 57–66. MR 721596, DOI 10.1007/BF01298934
John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054
John S. Wilson, The probability of generating a soluble subgroup of a finite group, J. Lond. Math. Soc. (2) 75 (2007), no. 2, 431–446. MR 2340237, DOI 10.1112/jlms/jdm020