Conjugation-invariantly permutably complemented subgroup

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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 $g\in G$.

Relation with other properties

Stronger properties

• Permutably complemented normal subgroup

Weaker properties

• Permutably complemented subgroup
• Lattice-complemented subgroup