# Institute of Mathematics

## Every $2$-group with all subgroups normal-by-finite is locally finite

#### Enrico Jabara

###### Received September 30, 2016.   First published February 14, 2018.

Abstract:  A group $G$ has all of its subgroups normal-by-finite if $H/H_G$ is finite for all subgroups $H$ of $G$. The Tarski-groups provide examples of $p$-groups ($p$ a "large" prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$-group with every subgroup normal-by-finite is locally finite. We also prove that if $| H/H_G | \leq2$ for every subgroup $H$ of $G$, then $G$ contains an Abelian subgroup of index at most $8$.
Keywords:  $2$-group; locally finite group; normal-by-finite subgroup; core-finite group
Classification MSC:  20F50, 20F14, 20D15
DOI:  10.21136/CMJ.2018.0504-16

PDF available at:  Institute of Mathematics CAS

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Affiliations:   Enrico Jabara, DFBC - Università di Venezia, Dorsoduro 3484/D - 30123 Venezia, Italy, e-mail: jabara@unive.it

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