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:
- It takes each normal subgroup to itself (bijectively).
- On each normal subgroup, it restricts to that subgroup as an automorphism of that subgroup.
- It descends to an automorphism of the quotient group for any quotient map.
Definition with symbols
An automorphism of a group is termed quotientable or normal or normal subgroup-preserving if it satisfies the following equivalent conditions:
- For any , .
- For any , the restriction of to defines an automorphism of .
- For any , descends to an automorphism of .
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 for all normal subgroups of is not sufficient for the automorphism to be a normal automorphism. For instance, the map 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
View other such automorphism properties
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
Stronger properties
- Inner automorphism
- Class-preserving automorphism
- Subgroup-conjugating automorphism
- Strong monomial automorphism: For full proof, refer: Strong monomial implies normal
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.
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.
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 groups^{More info}
- Normal automorphisms of free groups by Abraham S.T.-Lue, Journal of Algebra, 64, 1980, Page 52-53^{More info}
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