University of Isfahan International Journal of Group Theory 2251-7650 7 3 2018 09 01 Inertial properties in groups 17 62 21611 10.22108/ijgt.2017.21611 EN Ulderico Dardano Dipartimento Matematica e Appl., v. Cintia, M.S.Angelo 5a, I-80126 Napoli (Italy) Dikran Dikranjan Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy. Silvana Rinauro Silvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a della Basilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy. Journal Article 2016 12 01 ‎‎Let $G$ be a group and $p$ be an endomorphism of $G$‎. ‎A subgroup $H$ of $G$ is called $p$-inert if $H^p\cap H$ has finite index in the image $H^p$‎. ‎The subgroups that are $p$-inert for all inner automorphisms of $G$ are widely known and studied in the literature‎, ‎under the name inert subgroups‎. ‎The related notion of inertial endomorphism‎, ‎namely an endomorphism $p$ such that all subgroups of $G$ are $p$-inert‎, ‎was introduced in \cite{DR1} and thoroughly studied in \cite{DR2,DR4}‎. ‎The ``dual‎" ‎notion of fully inert subgroup‎, ‎namely a subgroup that is $p$-inert for all endomorphisms of an abelian group $A$‎, ‎was introduced in \cite{DGSV} and further studied in \cite{Ch+‎, ‎DSZ,GSZ}‎. ‎The goal of this paper is to give an overview of up-to-date known results‎, ‎as well as some new ones‎, ‎and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra‎. ‎We survey on classical and recent results on groups whose inner automorphisms are inertial‎. ‎Moreover‎, ‎we show how‎ ‎inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces‎, ‎and can be helpful for the computation of the algebraic entropy of continuous endomorphisms‎.

https://ijgt.ui.ac.ir/article_21611_00d5ab9d6cd65813b0631a40fa7db9fb.pdf