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]
|Get subgroup property lookup help |Get exploration suggestions
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

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Conjugation-invariantly permutably complemented subgroup, all facts related to Conjugation-invariantly permutably complemented subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 $g\in G$ .