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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 |

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:

• $H_0=H$
• $H_{\alpha }\underset{\_}{◃}{H}_{\alpha +1}$ (viz $H_{\alpha }$ is a normal subgroup of $H_{\alpha +1}$)
• If $\alpha$ is a limit ordinal, then $H_{\alpha }=⟨H_{\gamma }{⟩}_{\gamma <\alpha }$, 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

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes		If $H\le K\le G$ are groups such that $H$ is an ascendant subgroup of $K$ and $K$ is an ascendant subgroup of $G$, then $H$ is an ascendant subgroup of $G$.
trim subgroup property	Yes		Every group is ascendant in itself, and the trivial subgroup is ascendant in any group.