Groups without proper abnormal subgroups

L. A Kurdachenko 1 , 1 , A Russo 2 , 2 ,  and G Vincenzi 3 , 3
  • 1 Department of Algebra, Dnepropetrovsk National University, Dnepropetrvsk, Ukraine.
  • 2 Dipartimento di Matematica, Seconda Università di Napoli, Caserta, Italy.
  • 3 Dipartimento di Matematica e Informatica, Università di Salerno, Salerno, Italy.

Abstract

A subgroup H of a group G is said to be abnormal in G if g ∈ <H, Hg> for each element gG. It is well known that every locally nilpotent group has no proper abnormal subgroups, but it is an open question whether the converse holds. In this article we prove this conjecture for some classes of infinite groups. In particular, it is proved that an FC-nilpotent group without proper abnormal subgroups is hypercentral. Also groups with finitely many abnormal subgroups are considered.

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The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

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