Class-preserving 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]

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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 |

Origin

Origin of the concept

The concept of class-preserving automorphism first took explicit shape when it was observed that there are automorphisms of groups that take each element to within its conjugacy class but are not inner. That is because there may not be a single element that serves uniformly as a conjugating candidate.

Definition

Symbol-free definition

An automorphism of a group is termed a class-preserving automorphism or class automorphism if it takes each element to within its conjugacy class.

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed a class automorphism or class-preserving automorphism if for every $g$ in $G$, there exists an element $h$ such that $\sigma (g)=hgh^{-1}$. The choice of $h$ may depend on $g$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Inner automorphism	conjugation by an element	inner implies class-preserving	class-preserving not implies inner	|FULL LIST, MORE INFO
Locally inner automorphism	preserves conjugacy classes of finite tuples	locally inner implies class-preserving	class-preserving not implies locally inner	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
IA-automorphism	induces identity map on abelianization	class-preserving implies IA	IA not implies class-preserving	|FULL LIST, MORE INFO
Normal automorphism	sends each normal subgroup to itself	class-preserving implies normal	normal not implies class-preserving	|FULL LIST, MORE INFO
Weakly normal automorphism	sends each normal subgroup to a subgroup to itself	(via normal)	(via normal)	|FULL LIST, MORE INFO
Extended class-preserving automorphism	sends each element to conjugate or conjugate of inverse		extended class-preserving not implies class-preserving	|FULL LIST, MORE INFO
Rational class-preserving automorphism	sends each element to conjugate of element generating same cyclic group			|FULL LIST, MORE INFO
Center-fixing automorphism	fixes every element of center	class-preserving implies center-fixing	center-fixing not implies class-preserving	|FULL LIST, MORE INFO

Related properties

• Subgroup-conjugating automorphism: Further information: Class-preserving not implies subgroup-conjugating, Subgroup-conjugating not implies class-preserving
• Class-inverting automorphism

Facts

• Class-preserving automorphism group of finite p-group is p-group

Metaproperties

Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties

Clearly, a product of class automorphisms is a class automorphism, and the inverse of a class automorphism is a class automorphism. Thus, the class automorphisms form a group which sits as a subgroup of the automorphism group. Moreover, this subgroup contains the group of inner automorphisms, and is a normal subgroup inside the automorphism group.

Direct product-closedness

This automorphism property is direct product-closed
View a complete list of direct product-closed automorphism properties

Let $G_{1}$ and $G_{2}$ be groups and $\sigma _{1},\sigma _{2}$ be class automorphisms of $G_{1},G_{2}$ respectively. Then, $\sigma _{1}\times \sigma _{2}$ is a class automorphism of $G_{1}\times G_{2}$.

Here, $\sigma _{1}\times \sigma _{2}$ is the automorphism of $G_{1}\times G_{2}$ that acts as $\sigma _{1}$ on the first coordinate and $\sigma _{2}$ on the second.

References

• On the outer automorphisms of a group by William Burnside, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Volume 11, (Year 1913): ^{More info}
• Finite groups with class-preserving outer automorphisms by G. E. Wall, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Page 315 - 320(Year 1947): ^{More info}
• Class-preserving automorphisms of finite groups by Martin Hertweck, Journal of Algebra, Volume 241, Issue 1, 1 July 2001, Pages 1-26^{More info}
• Class-preserving automorphisms of finite p-groups by Manoj K. Yadav, Journal of the London Mathematical Society, 2007, Page 755-772^{More info}

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