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
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

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

• $H_0 = H$
• $H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}$ (viz $H_\alpha$ is a normal subgroup of $H_{\alpha + 1}$) for every ordinal $\alpha$.
• If $\alpha$ is a limit ordinal, then $H_\alpha = \bigcup_{\gamma < \alpha} H_\gamma$, i.e., it is the union of all preceding subgroups. Note that the union of any ascending chain of subgroups is indeed a sugbroup (in fact, more generally, directed union of subgroups is subgroup). We can also define $H_{\alpha}$as $\langle H_\gamma \rangle_{\gamma < \alpha}$, 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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup	we can get a series that reaches the group in one step.
subnormal subgroup	we can get a series that reaches the group in finitely many steps.	(obvious)	ascendant not implies subnormal	|FULL LIST, MORE INFO
hypernormalized subgroup	we can use the series where the subgroup for each successor ordinal is the normalizer in the whole group of the subgroup for the ordinal.	(obvious)		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
serial subgroup

Opposites

• Self-normalizing subgroup

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	ascendance is transitive	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.
intermediate subgroup condition	Yes	ascendance satisfies intermediate subgroup condition	If $H \le K \le G$ are groups such that $H$ is an ascendant subgroup of $G$, then $H$ is an ascendant subgroup of $K$.
strongly intersection-closed subgroup property	Yes	ascendance is strongly intersection-closed	If $H_i, i \in I$, are all ascendant subgroups of a group $G$, so is $\bigcap_{i \in I} H_i$.
transfer condition	Yes	ascendance satisfies transfer condition	If $H,K \le G$ are subgroups and $H$ is ascendant in $G$, then $H \cap K$ is ascendant in $K$.
image condition	Yes	ascendance satisfies image condition	Suppose $H$ is an ascendant subgroup of a group $G$, and $\varphi:G \to K$ is a surjective homomorphism of groups. Then, $\varphi(H)$ is an ascendant subgroup of $K$.