Permutably complemented subgroup

Symbol-free definition
A subgroup of a group is said to be permutably complemented if there is another subgroup that intersects it trivially and such that the product group of these groups is the whole group.

This other subgroup is termed a permutable complement.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be permutably complemented if there is a subgroup $$K$$ of $$G$$ such that $$HK = KH = G$$ and $$H \cap K$$ is trivial.

$$K$$ is termed a permutable complement to $$H$$.

Stronger properties

 * Weaker than::Retract
 * Weaker than::Direct factor
 * Weaker than::Permutably complemented normal subgroup
 * Weaker than::Strongly permutably complemented subgroup
 * Weaker than::Conjugation-invariantly permutably complemented subgroup

Weaker properties

 * Stronger than::Lattice-complemented subgroup

Metaproperties
The trivial subgroup and the whole group are permutable complements of each other, hence both are permutably complemented subgroups.

if $$H$$ is a permutably complemented subgroup of $$G$$, then $$H$$ is also permutably complemented in every intermediate subgroup $$K$$ of $$G$$. This follows from the modular property of groups.

If $$H$$ is a permutably complemented subgroup of $$K$$ and $$K$$ is a permutably complemented subgroup of $$G$$, $$H$$ need not be permutably complemented in $$G$$. The reason is that the permutable complement of $$H$$ in $$K$$ need not permute with the permutable complement of $$K$$ in $$G$$.

An intersection of permutably complemented subgroups need not be permutably complemented.

If $$H$$ is a permutably complemented subgroup of $$G$$, and $$K$$ is some subgroup of $$G$$, $$H \cap K$$ need not be permutably complemented in $$K$$.

Suppose $$H \le K \le G$$ are such that $$H$$ is a permutably complemented normal subgroup of $$G$$ and $$K/H$$ is a permutably complemented subgroup of $$G/H$$. Then, $$K$$ is also permutably complemented in $$G$$.

If $$\varphi:G \to K$$ is a surjective homomorphism, and $$H$$ is a permutably complemented subgroup of $$G$$, $$\varphi(H)$$ is a permutably complemented subgroup of $$K$$.