Normal automorphism

Symbol-free definition
An automorphism of a group is termed normal or quotientable or normal subgroup-preserving if it satisfies the following equivalent conditions:


 * 1) It takes each defining ingredient::normal subgroup to itself (bijectively).
 * 2) On each normal subgroup, it restricts to that subgroup as an automorphism of that subgroup.
 * 3) It descends to an automorphism of the defining ingredient::quotient group for any quotient map.

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is termed quotientable or normal or normal subgroup-preserving if it satisfies the following equivalent conditions:


 * 1) For any $$N \triangleleft G$$, $$\sigma(N) = N$$.
 * 2) For any $$N \triangleleft G$$, the restriction of $$\sigma$$ to $$N$$ defines an automorphism of $$N$$.
 * 3) For any $$N \triangleleft G$$, $$\sigma$$ descends to an automorphism of $$G/N$$.

Equivalence of definitions
The equivalence of definitions (1) and (2) follows from the fact that restriction of endomorphism to invariant subgroup is endomorphism. The equivalence with (3) is also straightforward.

Note that the condition $$\sigma(N) \subseteq N$$ for all normal subgroups $$N$$ of $$G$$ is not sufficient for the automorphism to be a normal automorphism. For instance, the map $$x \mapsto 2x$$ on the additive group of rational numbers sends each normal subgroup to within itself, but it is not a normal automorphism because there are normal subgroups to which its restriction is not bijective.

Formalisms
In the general language of a variety of algebras, the property of being a normal automorphism translates to the property of being an IC-automorphism: an automorphism that leaves every congruence invariant.

Stronger properties

 * Weaker than::Inner automorphism
 * Weaker than::Class-preserving automorphism
 * Weaker than::Subgroup-conjugating automorphism
 * Weaker than::Strong monomial automorphism:

Weaker properties

 * Stronger than::Weakly normal automorphism: A weakly normal automorphism is an automorphism that sends each normal subgroup to within itself; the restriction to each normal subgroup need not be bijective.

Related subgroup properties

 * Normal subgroup is the invariance property corresponding to normal automorphisms.
 * Transitively normal subgroup is the balanced subgroup property corresponding to normal automorphisms.