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Inertial properties in groups
‎‎Let be a group and be an endomorphism of ‎. ‎A subgroup of is called -inert if has finite index in the image ‎. ‎The subgroups that are -inert for all inner automorphisms of are widely known and studied in the literature‎, ‎under the name inert subgroups‎. ‎The related notion of inertial endomorphism‎, ‎namely an endomorphism such that all subgroups of are -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 -inert for all endomorphisms of an abelian group ‎, ‎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