On strongly autinertial groups

Abstract

A subgroup $ X $ of $ G $ is said to be inert under automorphisms (autinert) if $ |X : X^\alpha \cap X | $ is finite for all $ \alpha \in Aut(G)$ and it is called strongly autinert if $ |:X | $ is finite for all $ \alpha \in Aut(G).$ A group is called strongly autinertial if all subgroups are strongly autinert. In this article, the strongly autinertial groups are studied. We characterize such groups for a finitely generated case. Namely, we prove that a finitely generated group $ G $ is strongly autinertial if and only if one of the following hold:\vs{-2mm} \begin{itemize} \item[i)] $ G $ is finite;\vs{-2mm} \item[ii)] $ G= \langle a \rangle \ltimes F $ where $ F $ is a finite subgroup of $ G $ and $ \langle a \rangle $ is a torsion-free subgroup of $ G. $ \end{itemize}\vs{-2mm} Moreover, in the preliminary part, we give basic results on strongly autinert subgroups.