Conjugation-invariantly permutably complemented subgroup

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 $$g \in G$$.

Stronger properties

 * Permutably complemented normal subgroup

Weaker properties

 * Permutably complemented subgroup
 * Lattice-complemented subgroup