  The aim of the paper is to discuss the relation subgroups of the Grothendieck
groups of extriangulated categories and certain other subgroups. It is shown
that a locally finite extriangulated category $\C$ has Auslander-Reiten
$\E-$triangles and the relations of the Grothendieck group $K_{0}(\C)$ are
generated by the Auslander-Rieten $\E-$triangles. A partial converse result is
given when restricting to the triangulated categories with a cluster tilting
subcategory: in the triangulated category $\C$ with a cluster tilting
subcategory, the relations of the Grothendieck group $K_0(\C)$ are generated by
Auslander-Reiten triangles if and only if the triangulated category $\C$ is
locally finite. It is also shown that there is a one-to-one correspondence
between subgroups of $K_{0}(\C)$ containing the image of $\mathcal G$ and dense
$\mathcal G-$(co)resolving subcategories of $\C$ where $\mathcal G$ is a
generator of $\C,$ which generalizes results about classifying subcategories of
a triangulated \cite{t} or an exact category $\C$ \cite{m} by subgroups of
$K_{0}(\C)$.
