# Conjugation-invariantly permutably complemented 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]

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is said to be conjugation-invariantly permutably complemented if it satisfies both the conditions below:

• It is permutably complemented, viz the set of its permutable complements is empty
• The set of its permutable complements is closed under conjugation. In other words, any conjugate of a permutable complement is also a permutable complement.

### Definition with symbols

A subgroup $H$ of a group $G$ is said to be conjugation-invariantly permutably complemented if the following are true:

• There exists a subgroup $K$ of $G$ such that $HK = G$ and $H \cap K$ is trivial (in other words, a permutable complement of $H$ in $G$)
• If $K$ and $H$ are permutable complements, then $K^g$ and $H$ are also permutable complements for any $g \in G$.