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 |
- 1 Definition
- 2 Relation with other properties
- 3 Metaproperties
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
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
An ascendant subgroup of an ascendant subgroup is ascendant. The proof relies on simply concatenating the two ascending series.
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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Thetrivial subgroup and the whole group are both normal subgroups, hence they are also both ascendant subgroups.
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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Any ascendant subgroup of a group is also ascendant in every intermediate subgroup. The proof of this follows by intersecting every member of the ascending series with the intermediate subgroup and observing that normality at each stage is preserved.
YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness
An arbitrary intersection of ascendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.