Normal subgroup of group of prime power order

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 $$G$$.

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:


 * 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.

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.