This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |
This is a variation of subnormality|Find other variations of subnormality |
- (viz is a normal subgroup of ) for every ordinal .
- If is a limit ordinal, then .
and such that there is some ordinal such that .
In terms of the descendant closure operator
Relation with other properties
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|transitive subgroup property||Yes||descendance is transitive||If are groups such that is a descendant subgroup of and is a descendant subgroup of , then is a descendant subgroup of .|
|trim subgroup property||Yes||Every group is descendant in itself, and the trivial subgroup is descendant in any group.|
|intermediate subgroup condition||Yes||descendance satisfies intermediate subgroup condition||If are groups such that is descendant in , then is descendant in .|
|strongly intersection-closed subgroup property||Yes||descendance is strongly intersection-closed||If , are all descendant subgroups of , so is the intersection .|
|image condition||No||descendance does not satisfy image condition||It is possible to have groups and , a descendant subgroup of and a surjective homomorphism such that is not a descendant subgroup of .|