# Normal p-complement

From Groupprops

The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter

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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 -subgroup) as a permutable complement.

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

### Definition with symbols

A subgroup of a group is said to be a normal -complement if it satisfies the following equivalent conditions:

- is normal in and there is a Sylow -subgroup of such that and is trivial.
- is normal in and for
*every*Sylow -subgroup of , and is trivial. - is a normal Hall subgroup of whose order is relatively prime to and whose index is a power of . In other words, is a normal -Hall subgroup of .

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

## Facts

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

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