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THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 99 / Issue 1 / August 2015
- Published online by Cambridge University Press:
- 19 January 2015, pp. 108-127
- Print publication:
- August 2015
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- Article
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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.
