Normal-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 normal-monotone if whenever $N\triangleleft G$ is a normal subgroup, then $f(N)\leq f(G)$.

Relation with other properties

Stronger properties

• Verbal subgroup-defining function
• Monotone subgroup-defining function

Weaker properties

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

Subgroup-defining functions satisfying this property

Frattini subgroup

The Frattini subgroup is not a monotone subgroup-defining function, but it is normal-monotone. The crucial fact used here is the modular property of groups. For full proof, refer: Frattini subgroup is normal-monotone