# 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]
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:

• $H_0 = H$
• $H_\alpha \ \underline{\triangleleft} \ H_{\alpha + 1}$ (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.

## 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.