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A group has all of its subgroups normal-by-finite if is finite for all subgroups of . The Tarski-groups provide examples of -groups ( 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 -group with every subgroup normal-by-finite is locally finite. We also prove that if for every subgroup of , then contains an Abelian subgroup of index at most .
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