Monotone subgroup-defining function

Definition with symbols
A subgroup-defining function $$f$$ is said to be monotone if whenever $$N \le G$$ is a subgroup, then $$f(N) \le f(G)$$.

Stronger properties

 * Verbal subgroup-defining function

Weaker properties

 * Normal-monotone subgroup-defining function
 * Characteristic-monotone subgroup-defining function
 * Direct factor-monotone subgroup-defining function

Commutator subgroup
The commutator subgroup of any subgroup is contained in the commutator subgroup of the whole group. This follows from the fact that any commutator of two elements in the subgroup is also a commutator of two elements in the whole group.