On strongly autinertial groups

Abstract

A subgroup X of G is said to be inert under automorphisms (autinert) if |X : X α ∩ X| is finite for all α ∈ Aut(G) and it is called strongly autinert if | < X, X α >: X| is finite for all α ∈ 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:where F is a finite subgroup of G and ⟨a⟩ is a torsion-free subgroup of G.Moreover, in the preliminary part, we give basic results on strongly autinert subgroups.