Czechoslovak Mathematical Journal, first online, pp. 1-6
Every $2$-group with all subgroups normal-by-finite is locally finite
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
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