Suppose $G$ is a finitely generated group. What conditions on $G$(or some subgroups of $G$) will force it to be a Noetherian group?
Of course, if all subgroups are finitely generated or ACC on subgroups is satisfied, then we are done by definition. Are there any known results which can help prove neotherianity of group $G$, if $G$ was known to be finitely generated?
For a solvable group being Noetherian is equivalent with being polycyclic. Similarly, a finitely generated nilpotent group is Noetherian. You also have the "standard subgroup-quotient-extension-closeness", i.e. subgroups of Noetherian groups are Noetherian, quotients of Noetherian groups are Noetherian and if a normal subgroup and the corresponding quotient is Noetherian then so is the group. If the the group ring $R[G]$ is noetherian (in the sense of rings) for some non-zero commutative ring $R$ then $G$ is Noetherian. Moreover Noetherian groups need to be Hopfian (necessary condition) and it is (to my knowledge) open if "being Noetherian" is a quasi-isometric invariant of finitely presented groups.