Normal subgroup of finite group

Definition
A subgroup $$H$$ of a group $$G$$ is termed a normal subgroup of finite group if it satisfies the following equivalent conditions:


 * 1) The whole group $$G$$ is a finite group and the subgroup $$H$$ is a normal subgroup of it.
 * 2) There exists a finite group $$K$$ containing $$G$$ such that $$H$$ is a characteristic subgroup of $$K$$.
 * 3) $$G$$ is a finite group and there is a generating set $$A$$ for $$H$$ and a generating set $$B$$ for $$G$$ such that the conjugate of any element of $$A$$ by any element of $$B$$ is in $$H$$.

Equivalence of definitions

 * The equivalence of definitions (1) and (2) follows from the finite NPC theorem.

Examples
See normal subgroup.

How it differs from normality in general
The property of normality inside a finite group is largely similar to the property of normality in an arbitrary group -- most of the property implications, metaproperties, etc. hold for finite groups. There are some small differences:


 * To check that a subgroup of a finite group is normal, it suffices to check that it is sent to within itself under all elements in a generating set of the whole group. For an infinite group, we need to check that it is sent to within itself by all elements of a generating set and their inverses.
 * Normal subgroups of finite groups, and more generally finite normal subgroups, behave nicely with respect to amalgams: finite normal implies amalgam-characteristic. This is not true for normal subgroups in arbitrary groups.