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 |
A subgroup of a group is said to be ascendant if there is an ascending series of subgroups indexed by ordinals, each normal in the next, that starts at the given subgroup, and terminates at the whole group.
Definition with symbols
- (viz is a normal subgroup of )
- If is a limit ordinal, then , i.e., it is the join of all preceding subgroups.
and such that there is some ordinal such that .
In terms of the ascendant closure operator
Relation with other properties
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|transitive subgroup property||Yes||If are groups such that is an ascendant subgroup of and is an ascendant subgroup of , then is an ascendant subgroup of .|
|trim subgroup property||Yes||Every group is ascendant in itself, and the trivial subgroup is ascendant in any group.|