For any integer n \neq 0, 1, a group G is said to be n-abelian if it satisfies the identity (xy)^n = x^n y^n. More generally, G is called an Alperin group if it is n-abelian for some n \neq 0, 1. We consider two natural ways to generalize the concept of n-abelian group: the former leads to define n-soluble and n-nilpotent groups, the latter to define n-Levi and n-Bell groups. The main goal of this paper is to present classes of generalized n-abelian groups and to point out connections among them. Besides, Section 5 contains unpublished combinatorial characterizations for Bell groups and for Alperin groups. Finally, in Section 6 we mention results of arithmetic nature.
On n-abelian groups and their generalizations
TORTORA, ANTONIO
2011
Abstract
For any integer n \neq 0, 1, a group G is said to be n-abelian if it satisfies the identity (xy)^n = x^n y^n. More generally, G is called an Alperin group if it is n-abelian for some n \neq 0, 1. We consider two natural ways to generalize the concept of n-abelian group: the former leads to define n-soluble and n-nilpotent groups, the latter to define n-Levi and n-Bell groups. The main goal of this paper is to present classes of generalized n-abelian groups and to point out connections among them. Besides, Section 5 contains unpublished combinatorial characterizations for Bell groups and for Alperin groups. Finally, in Section 6 we mention results of arithmetic nature.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
