# Normal subgroup of group of prime power order

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property imposed on theambient group: group of prime power order

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

Suppose is a group and is a subgroup of . We say that is a **normal subgroup of group of prime power order** if is a group of prime power order and is a normal subgroup of .

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups whose group part dissatisfies property group of prime power orderVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

### Extreme examples

- The trivial group is always a normal subgroup.
- Every group of prime power order is normal as a subgroup of itself.

### Lots of normal subgroups

- In an abelian group of prime power order, or in a Hamiltonian group (which is a direct product of an elementary abelian 2-group and the quaternion group) every subgroup is normal.
- There exists a normal subgroup of every order dividing the order of the group. Moreover, there is a congruence condition on number of subgroups of given prime power order, which guarantees that the number of normal subgroups is congruent to modulo .
- Every normal subgroup contains normal subgroups of the whole group of every order dividing its order, and every normal subgroup is contained in normal subgroups of the whole group of every order that's a multiple of its order and divides the order of the group.
- Every maximal subgroup is normal.

### Normal subgroups arising from subgroup-defining functions

Subgroup-defining functions, such as characteristic p-functors, give rise to normal subgroups. Examples include:

- The derived subgroup and members of the derived series and lower central series.
- The center and member of the upper central series.
- The Frattini subgroup and members of the Frattini series.
- The omega subgroups and agemo subgroups.

### Particular examples

Group | Order | Normal subgroups | Non-normal subgroups |
---|---|---|---|

dihedral group:D8 (subgroup structure) | cyclic maximal subgroup of dihedral group:D8, Klein four-subgroups of dihedral group:D8, center of dihedral group:D8, whole group, trivial subgroup | 2-subnormal subgroups of order two generated by non-central elements | |

quaternion group (subgroup structure) | cyclic maximal subgroups of quaternion group, center of quaternion group, whole group, trivial subgroup | none | |

prime-cube order group:U(3,p) (subgroup structure) | center, elementary abelian normal subgroups of order , whole group, trivial subgroup | non-central subgroups of order | |

semidirect product of cyclic group of prime-square order and cyclic group of prime order (subgroup structure) | center, elementary abelian normal subgroup and cyclic normal subgroups of order , whole group, trivial subgroup | non-central subgroups of order |

## Relation with other properties

### How it differs from normality in general

All the metaproperties that normality satisfies continue to be satisfied in the context of groups of prime power order. Also, normality is not transitive in general and also when restricted to groups of prime power order -- see normality is not transitive for any nontrivially satisfied extension-closed group property.

On the other hand, there are some additional things we can say in the context of groups of prime power order:

- Prime power order implies center is normality-large: Every nontrivial normal subgroup intersects the center nontrivially.
- There exist normal subgroups of every order, and there are also various replacement theorems (see Category:Replacement theorems) that allow one to replace subgroups of certain kinds by similar-looking normal subgroups.

### Stronger properties

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Normal subgroup of nilpotent group | |FULL LIST, MORE INFO | |||

Normal subgroup contained in upper central series member | |FULL LIST, MORE INFO | |||

Normal subgroup contained in the hypercenter | Normal subgroup of nilpotent group|FULL LIST, MORE INFO |