Class-preserving automorphism

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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 |

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. In 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	Hall-extensible automorphism, Locally inner automorphism|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
automorphism that preserves conjugacy classes for a generating set	there exists a generating set all of whose elements are sent to within their conjugacy class by the automorphism	(obvious)	preserves conjugacy classes for a generating set not implies class-preserving	|FULL LIST, MORE INFO
IA-automorphism	induces identity map on abelianization	class-preserving implies IA	IA not implies class-preserving	Automorphism that preserves conjugacy classes for a generating set|FULL LIST, MORE INFO
normal automorphism	sends each normal subgroup to itself	class-preserving implies normal	normal not implies class-preserving	Extended class-preserving automorphism, Rational class-preserving automorphism|FULL LIST, MORE INFO
weakly normal automorphism	sends each normal subgroup to a subgroup of itself	(via normal)	(via normal)	Extended class-preserving automorphism, Normal automorphism, Rational class-preserving automorphism|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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