  Let G be a group and f be an endomorphism of G. A subgroup H of G is called
f-inert if the meet of Hf and H has finite index in the image Hf. The subgroups
that are f-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 f such that all subgroups of G
are f-inert, was introduced in [30] and thoroughly studied in [31, 33]. The
dual notion of fully inert subgroup, namely a subgroup that is f-inert for all
endomorphisms of an abelian group A, was introduced in [42] and further studied
in [43, 46, 65, 21]. 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 automorphism 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.
