Noetherian implies Hopfian

Statement
Any Noetherian group (i.e., a group in which every subgroup is finitely generated) is Hopfian (i.e., it is not isomorphic to the quotient by any nontrivial subgroup).

Noetherian group
A group is termed Noetherian if every subgroup of the group is finitely generated. Equivalently, it satisfies the ascending chain condition of subgroups: every ascending chain of subgroups stabilizes after a finite length.

Hopfian group
A group is termed Hopfian if it is not isomorphic to its quotient by any nontrivial normal subgroup.

Facts used

 * 1) uses::Noetherian implies ascending chain condition on normal subgroups
 * 2) uses::Ascending chain condition on normal subgroups implies Hopfian

Proof
The proof follows by piecing together facts (1) and (2).