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 with symbols
A subgroup-defining function is said to be monotone if whenever is a subgroup, then .
Relation with other properties
- Normal-monotone subgroup-defining function
- Characteristic-monotone subgroup-defining function
- Direct factor-monotone subgroup-defining function
Subgroup-defining functions satisfying this property
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.