# Difference between revisions of "Normal p-complement"

The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
View other prime-parametrized subgroup properties | View all subgroup properties

## 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: