Question:Direct factor complement

Q: If $$H$$ is a subgroup of $$G$$ and has a permutable complement $$K$$ (i.e., the product of subgroups $$HK$$ equals $$G$$ and $$H \cap K$$ is trivial), then is $$H$$ a direct factor of $$G$$?

A: Not in general.

To qualify for an answer references::internal direct product, we need an additional condition: both $$H$$ and $$K$$ should be normal subgroups.


 * The existence of a permutable complement $$K$$ makes $$H$$ a answer references::permutably complemented subgroup.
 * Further, if $$H$$ is a normal subgroup, it becomes a answer references::complemented normal subgroup and $$G$$ becomes an answer references::internal semidirect product of $$H$$ by $$K$$. $$K$$ is in this case a answer references::retract and we say that it has a answer references::normal complement $$H$$.
 * Similar situation, with roles of $$H$$ and $$K$$ interchanged, if $$K$$ is the normal subgroup.

See also permutably complemented not implies normal, complemented normal not implies direct factor, retract not implies direct factor.