The number of profinite groups with a specified Sylow subgrou

TL;DR

This paper explores the classification of profinite groups with a given Sylow subgroup, revealing conditions under which there are finitely or infinitely many such groups up to isomorphism.

Contribution

It provides new insights into the structure and classification of profinite groups with specified Sylow subgroups, including examples and finiteness results.

Findings

Existence of infinite ascending chains of soluble groups in certain cases

Finiteness of isomorphism classes when the Sylow subgroup is just infinite

Characterization of conditions affecting the number of such groups

Abstract

Let $S$ be a finitely generated pro-$p$ group. Let $\Emb(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects non-trivially with every non-trivial normal subgroup of $G$. In this paper, we investigate the question of whether or not $\Emb(S)$ has finitely many isomorphism classes. For instance, we give an example where $\Emb(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $\Emb(S)$ contains only finitely many isomorphism classes in the case that $S$ is just infinite.