Groups in which every element has a paracentralizer of finite index

Abstract

A group G is called an FP-group if for each element g of G there exists a subgroup X_g of G such that the index $|G:X_g|$ is finite and $Y^g=Y$ for all subgroups Y of X_g, that is, if every element of G induces by conjugation a power automorphism on a suitable subgroup of finite index of G. Thus groups with finite conjugacy classes have the FP-property, and the aim of this article is to study the behavior of FP-groups in relation to the known theory of FC-groups.

Communicated by Sudarshan Kumar Sehgal

Keywords:

• FC-group
• FP-group
• paracentralizer
• power automorphism

Mathematics Subject Classification (2010):

• 20F24

Acknowledgments

The second author is grateful to the University of Napoli Federico II for its hospitality during the preparation of this article.

Additional information

Funding

The first, the third and the fourth author are supported by GNSAGA (INdAM) and are members of AGTA – Advances in Group Theory and Applications (www.advgrouptheory.com).