Ulderico Dardano - Dipartimento Matematica e Appl., v. Cintia, M.S.Angelo ۵a, I-۸۰۱۲۶ Napoli (Italy)
Dikran Dikranjan - Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze ۲۰۶, ۳۳۱۰۰ Udine, Italy.
Silvana Rinauro - Silvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a della Basilicata, Via dell’Ateneo Lucano ۱۰, I-۸۵۱۰۰ Potenza, Italy.

‎‎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{DR۱} and thoroughly studied in \cite{DR۲,DR۴}‎. ‎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.

‎‎commensurable‎, ‎inert‎, ‎inertial endomorphism‎, ‎entropy‎, ‎intrinsic entropy‎, ‎scale function‎, ‎growth‎, ‎locally compact group‎, ‎locally linearly compact space‎, ‎Mahler measure‎, ‎Lehmer problem