  We carry out a study of groups $G$ in which the index of any infinite
subgroup is finite. We call them restricted-finite groups and characterize
finitely generated not torsion restricted-finite groups. We show that every
infinite restricted-finite abelian group is isomorphic to $\mathbb{Z}\times K$
or $\mathbb{Z}_{p^\infty}\times K$, where $K$ is a finite group and $p$ is a
prime number. We also prove that a group $G$ is infinitely generated
restricted-finite if and only if $G = AT$, where $A$ and $T$ are subgroups of
$G$ such that $A$ is normal quasicyclic and $T$ is finite. As an application of
our results, we show that if $G$ is not torsion with finite $G'$ and the
group-ring $RG$ has restricted minimum condition then $R$ is a semisimple ring
and $G\cong T\rtimes\mathbb{Z} $, where $T$ is finite whose order is unit in
$R$. The converse is also true with certain conditions including $G = T\times
\mathbb{Z}
