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.
Jabara, Enrico
2018-01-01
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) | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.
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