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Keywords:
$2$-group; locally finite group; normal-by-finite subgroup; core-finite group
Summary:
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 $| H/H_{G} | \leq 2$ for every subgroup $H$ of $G$, then $G$ contains an Abelian subgroup of index at most $8$.
References:
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