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THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

Published online by Cambridge University Press:  19 January 2015

COLIN D. REID*
Affiliation:
University of Newcastle, School of Mathematical and Physical Sciences, University Drive, Callaghan NSW 2308, Australia email colin@reidit.net
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Abstract

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Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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