# Normal automorphism

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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This is a variation of inner automorphism|Find other variations of inner automorphism |

## Definition

### 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 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 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

### Variety formalism

This automorphism property can be described in the language of universal algebra, viewing groups as a variety of algebras
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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.

## Relation with other properties

### Weaker properties

• 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.

## References

• Normal automorphisms of free groups by Alexander Lubotzky, Journal of Algebra, 63, 1980, Page 494-498: This paper introduces the notion of normal automorphism and studies the relation between normal automorphisms and inner automorphisms, for free groupsMore info
• Normal automorphisms of free groups by Abraham S.T.-Lue, Journal of Algebra, 64, 1980, Page 52-53More info