# 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]
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.

## 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 $K$.