Permutable complements

Symbol-free definition
Two subgroup of a group are said to be permutable complements if:


 * Their intersection is trivial
 * Their product is the whole group

Definition with symbols
Two subgroups $$H$$ and $$K$$ of a group $$G$$ are termed permutable complements if the following two conditions hold:


 * $$H \cap K$$ is the trivial group
 * $$HK = G$$

Permutable complements need not be unique
Given a subgroup $$H$$ of $$G$$, there may or may not exist permutable complements of $$H$$. Moreover, there may exist multiple possibilities for a complement to $$H$$, and the multiple possibilities may not even be pairwise isomorphic.

For a normal subgroup, they are fixed upto isomorphism
Interestingly, when a subgroup is normal, then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the quotient group.

Other related facts

 * There may be multiple subgroups that are pairwise permutable complements
 * Retract not implies every permutable complement is normal
 * Permutable complement to normal subgroup is isomorphic to quotient
 * Permutable complements to abelian normal subgroup are automorphic
 * Retract not implies normal complements are isomorphic
 * Permutable complements to normal subgroup need not be automorphic