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arXiv · 1511.08760

On small profinite groups

Patrick Helbig
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Abstract

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if every subgroup of finite index is open. We show that two profinite groups that are elementarily equivalent, in the first-order language of groups, are isomorphic if one of them is strongly complete, extending a result of Moshe Jarden and Alexander Lubotzky which treats the case of finitely generated profinite groups.

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Patrick Helbig. 2015-12-28. On small profinite groups. https://arxiv.org/abs/1511.08760

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