Ascending chain condition on subnormal subgroups is normal subgroup-closed

Property-theoretic statement
The property of being a group satisfying ascending chain condition on subnormal subgroups is a normal subgroup-closed group property.

Statement with symbols
Suppose $$G$$ is a group satisfying ascending chain condition on subnormal subgroups and $$N$$ is a normal subgroup of $$G$$. Then, $$N$$ is also a group satisfying ascending chain condition on subnormal subgroups.

Applications

 * Ascending chain condition on subnormal subgroups implies subnormal join property