Ascendant subgroup

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]

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If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
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Definition

Symbol-free definition

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

A subgroup $H$ of a group $G$ is termed ascendant if we have a series $H_\alpha$ for every ordinal $\alpha$ such that:

• $H_0 = H$
• $H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}$ (viz $H_\alpha$ is a normal subgroup of $H_{\alpha + 1}$)
• If $\alpha$ is a limit ordinal, then $H_\alpha = \langle H_\gamma \rangle_{\gamma < \alpha}$, i.e., it is the join of all preceding subgroups.

and such that there is some ordinal $\beta$ such that $H_\beta = G$.

In terms of the ascendant closure operator

The subgroup property of being an ascendant subgroup is obtained by applying the ascendant closure operator to the subgroup property of being normal.

Relation with other properties

Stronger properties

• Normal subgroup
• Subnormal subgroup
• Hypernormalized subgroup

Weaker properties

• Serial subgroup

Opposites

• Self-normalizing subgroup

Metaproperties

Transitivity

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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An ascendant subgroup of an ascendant subgroup is ascendant. The proof relies on simply concatenating the two ascending series.

Trimness

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.

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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An arbitrary intersection of ascendant subgroups is ascendant. This again follows by intersecting the ascending series for each subgroup.