On cyclic automorphisms of a group

Abstract

<jats:p> An endomorphism [Formula: see text] of a group [Formula: see text] is called a cyclic endomorphism if the subgroup [Formula: see text] is cyclic for all elements [Formula: see text] of [Formula: see text]. It can be proved that every cyclic endomorphism is normal, i.e. it commutes with every inner automorphism of [Formula: see text] (see [F. de Giovanni, M. L. Newell and A. Russo, On a class of normal endomorphisms of groups, J. Algebra its Appl. 13 (2014) 6pp.]). In this paper, some further properties of cyclic endomorphisms will be pointed out. Moreover, the structure of a group [Formula: see text] in which the group [Formula: see text] of cyclic automorphisms has finite index in [Formula: see text] will be investigated. </jats:p>

Generalized norms of groups: retrospective review and current status

“…M. Brescia and A. Russo [19] studied the cyclic norm C(G) of a group G, which is the intersection of the normalizers of every maximal locally cyclic subgroup of G. C(G) coincides with the set of all the elements of G including cyclic automorphisms on G. The quotient group C(G)/Z(G) is isomorphic with Inn(G)/CAut(G), where CAut(G) is the group of cyclic automorphisms of a group G. The authors proved that if C(G) has őnite index in G, then G is central-by-őnite.…”

On the Pronorm of a Group

“…Clearly, we may suppose that G is finitely generated. As G is a T-group, the factor group G/G (2) is either abelian or finite. In particular, G/G (2) has finite rank.…”

“…As G is a T-group, the factor group G/G (2) is either abelian or finite. In particular, G/G (2) has finite rank. Now let G/N be a finite quotient of G. Then G (2) ≤ N, since G/N is a T-group.…”

“…In particular, G/G (2) has finite rank. Now let G/N be a finite quotient of G. Then G (2) ≤ N, since G/N is a T-group. It follows that there is an upper bound for the (Prüfer) ranks of the finite quotients of G. Moreover, G is HNN-free so that it is polycyclic by [14,Theorem B].…”

“…Let G be a group whose norm has finite index. It follows that every subgroup of G is almost normal (that is, it has finitely many conjugates) and a famous theorem of Neumann [10] shows G is central-by-finite (see also [2]). In particular, the commutator subgroup of G is finite.…”