  A subgroup $H$ of a free group $F$ is called inert in $F$ if for every $G <
F$ the rank of the intersection of $H$ with $G$ is no grater than the rank of
$G$. In this paper we expand the known families of inert subgroups. We show
that the inertia property holds for 1-generator endomorphisms. Equivalently,
echelon subgroups in free groups are inert. An echelon subgroup is defined
through a set of generators that are in echelon form with respect to some
ordered basis of the free group, and may be seen as a generalization of a free
factor. For example, the fixed subgroups of automorphisms of finitely generated
free groups are echelon subgroups. The proofs follow mostly a graph-theoretic
or combinatorial approach.
