Question:Direct factor complement

This question is about direct factor, permutable complements| See more questions about direct factor

This question has type complete description conjecture

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 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 permutably complemented subgroup.
• Further, if $H$ is a normal subgroup, it becomes a complemented normal subgroup and $G$ becomes an internal semidirect product of $H$ by $K$. $K$ is in this case a retract and we say that it has a 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.