Descendant 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 RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
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

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

• $H_0 = G$
• $H_{\alpha + 1} \ \underline{\triangleleft} \ H_\alpha$ (i.e., $H_{\alpha + 1}$ is a normal subgroup of $H_\alpha$) for every ordinal $\alpha$.
• If $\alpha$ is a limit ordinal, then $H_\alpha = \bigcap_{\gamma < \alpha} H_\gamma$.

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

In terms of the descendant closure operator

The subgroup property of being an descendant subgroup is obtained by applying the descendant 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

Related properties

• Ascendant subgroup

Opposites

• Contranormal subgroup

Facts

Descendant-contranormal factorization

This result states that given any subgroup $H$ of $G$, there is a unique subgroup $K$ containing $H$ such that $H$ is contranormal in $K$ and $K$ is descendant in $G$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	descendance is transitive	If $H \le K \le G$ are groups such that $H$ is a descendant subgroup of $K$ and $K$ is a descendant subgroup of $G$, then $H$ is a descendant subgroup of $G$.
trim subgroup property	Yes		Every group is descendant in itself, and the trivial subgroup is descendant in any group.
intermediate subgroup condition	Yes	descendance satisfies intermediate subgroup condition	If $H \le K \le G$ are groups such that $H$ is descendant in $G$, then $H$ is descendant in $K$.
strongly intersection-closed subgroup property	Yes	descendance is strongly intersection-closed	If $H_i, i \in I$, are all descendant subgroups of $G$, so is the intersection $\bigcap_{i \in I} H_i$.
image condition	No	descendance does not satisfy image condition	It is possible to have groups $G$ and $K$, a descendant subgroup $H$ of $G$ and a surjective homomorphism $\varphi:G \to K$ such that $\varphi(H)$ is not a descendant subgroup of $K$.