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<p>A subgroup <i>X</i> of a group <i>G</i> is closed in the profinite topology if it can be obtained as intersection of a collection of subgroups of finite index of <i>G</i>, and a group <i>G</i> is said to be an <i>ERF</i>-group if all its subgroups are closed. It is proved here that if all large subgroups of an uncountable group <i>G</i> are closed, then <i>G</i> is an <i>ERF</i>-group, provided that either <i>G</i> is nilpotent-by-finite or it has finite conjugacy classes. Moreover, uncountable groups in which every large proper subgroup is an <i>ERF</i>-group are considered.</p>

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Uncountable extended residually finite groups

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月30日

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  • Maria De Falco,
  • Carmela Musella,
  • Antonella Zaccardo

摘要

A subgroup X of a group G is closed in the profinite topology if it can be obtained as intersection of a collection of subgroups of finite index of G, and a group G is said to be an ERF-group if all its subgroups are closed. It is proved here that if all large subgroups of an uncountable group G are closed, then G is an ERF-group, provided that either G is nilpotent-by-finite or it has finite conjugacy classes. Moreover, uncountable groups in which every large proper subgroup is an ERF-group are considered.