# Difference between revisions of "Normal p-complement"

From Groupprops

(→Facts) |
|||

Line 20: | Line 20: | ||

==Facts== | ==Facts== | ||

− | Normal <math>p</math>-complements may not always exist | + | Normal <math>p</math>-complements may not always exist. |

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

[[:Category:Normal p-complement theorems]] | [[:Category:Normal p-complement theorems]] |

## Latest revision as of 22:45, 1 July 2017

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 -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.

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