Monotone subgroup-defining function

This article defines a property of subgroup-defining functions, viz., a property that any subgroup-defining function may either satisfy or not satisfy

Definition

Definition with symbols

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

Relation with other properties

Stronger properties

• Verbal subgroup-defining function

Weaker properties

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

Subgroup-defining functions satisfying this property

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.