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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 | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.

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