Normal p-complement

Symbol-free definition
A subgroup of a group is said to be a normal p-complement if it is a normal subgroup and it has a Sylow subgroup (particularly a Sylow $$p$$-subgroup) as a permutable complement.

A group that has a normal $$p$$-complement is termed a p-nilpotent group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be a normal $$p$$-complement if it satisfies the following equivalent conditions:


 * 1) $$H$$ is normal in $$G$$ and there is a Sylow $$p$$-subgroup $$P$$ of $$G$$ such that $$HP = G$$ and $$H \cap P$$ is trivial.
 * 2) $$H$$ is normal in $$G$$ and for every Sylow $$p$$-subgroup $$P$$ of $$G$$, $$HP = G$$ and $$H \cap P$$ is trivial.
 * 3) $$H$$ is a normal Hall subgroup of $$G$$ whose order is relatively prime to $$p$$ and whose index is a power of $$p$$. In other words, $$H$$ is a normal $$p'$$-Hall subgroup of $$G$$.

If $$G$$ contains a normal $$p$$-complement $$H$$, we say that $$G$$ is a p-nilpotent group.

Facts
Normal $$p$$-complements may not always exist. Hall's theorem tells us that normal $$p$$-complements exist for all $$p$$ if and only if the group is solvable.

A complete list of normal p-complement theorems is available at:

Category:Normal p-complement theorems