# Normal subgroup of group of prime power order

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property imposed on the ambient 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

## Definition

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

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups whose group part dissatisfies property group of prime power order
VIEW: 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 $1$ modulo $p$.
• 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:

### Particular examples

Group Order Normal subgroups Non-normal subgroups
dihedral group:D8 (subgroup structure) $8 = 2^3$ 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) $8 = 2^3$ cyclic maximal subgroups of quaternion group, center of quaternion group, whole group, trivial subgroup none
prime-cube order group:U(3,p) (subgroup structure) $p^3$ center, elementary abelian normal subgroups of order $p^2$, whole group, trivial subgroup non-central subgroups of order $p$
semidirect product of cyclic group of prime-square order and cyclic group of prime order (subgroup structure) $p^3$ center, elementary abelian normal subgroup and cyclic normal subgroups of order $p^2$, whole group, trivial subgroup non-central subgroups of order $p$

## 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:

### Weaker properties

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