Normal p-complement

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Definition

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.

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

Category:Normal p-complement theorems