Topological groups with many small subgroups

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• 作者：Dikran Dikranjan^a ; ¹ ; ^{dikranja@dimi.uniud.it" class="auth_mail" title="E-mail the corresponding author} ; Dmitri Shakhmatov^b ; ² ; ^{dmitri.shakhmatov@ehime-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：18B30 ; 22A05 ; 22A10 ; 22A25 ; 54H11
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：200
• 期：Complete
• 页码：101-132
• 全文大小：712 K

文摘

We introduce and study a functorial topology on every group G having as a base the family of all subgroups of G. Making use of this topology, we obtain an equivalent description of the small subgroup generating property introduced by Gould [26]; see also Comfort and Gould [6]. This property implies minimal almost periodicity. Answering questions of Comfort and Gould [6], we show that every abelian group of infinite divisible rank admits a group topology having the small subgroup generating property. For unbounded abelian groups of finite divisible rank, we find a new necessary condition for the existence of a group topology having the small subgroup generating property, and we conjecture that this condition is also sufficient.