Abstract
Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup ⟨x,xα⟩ is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G/Z(G). In this paper, the class FW of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated.
Highlights
Following the work in [1], an automorphism α of G is called a cyclic automorphism if the subgroup h x, x α i is cyclic for every element x of G
In [1], it was proved that any cyclic automorphism of a group G is central, i.e., it acts trivially on the factor group G/Z ( G )
Recall here that a group G is said to be an FP-group if every element of G induces by conjugation a power automorphism on some subgroup of finite index of G
Summary
We will say that an element g of a group G induces by conjugation a weakly cyclic automorphism of G if there exists a normal subgroup W ( g) of G such that the index | G : W ( g)| is finite and the subgroup h x, x g i is cyclic for each element x of. In the first part of the article, the class F W of groups in which every element induces by conjugation a weakly cyclic automorphism will be investigated. Recall here that a group G is said to be an FP-group if every element of G induces by conjugation a power automorphism on some subgroup of finite index of G. Most of our notation is standard and can be found in [10]
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