Class-preserving automorphism: Difference between revisions

Latest revision as of 14:42, 1 September 2023

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 |

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	\|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	\|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 of 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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@@ Line 2: / Line 2: @@
 {{automorphism property}}
 {{variationof|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 automorphism|inner]]. That is because there may not be a <em>single</em> element that serves uniformly as a ''conjugating candidate''.
 ==Definition==
-===Symbol-free definition===
+An [[defining ingredient::automorphism]] of a group is termed a '''class-preserving automorphism''' or '''class automorphism''' if it takes each element to within its [[defining ingredient::conjugacy class]]. In symbols, an automorphism <math>\sigma</math> of a group <math>G</math> is termed a '''class automorphism''' or '''class-preserving automorphism''' if for every <math>g</math> in <math>G</math>, there exists an element <math>h</math> such that <math>\sigma(g) = hgh^{-1}</math>. The choice of <math>h</math> may depend on <math>g</math>.
-An [[defining ingredient::automorphism]] of a group is termed a '''class-preserving automorphism''' or '''class automorphism''' if it takes each element to within its [[defining ingredient::conjugacy class]].
-===Definition with symbols===
-An automorphism <math>\sigma</math> of a group <math>G</math> is termed a '''class automorphism''' or '''class-preserving automorphism''' if for every <math>g</math> in <math>G</math>, there exists an element <math>h</math> such that <math>\sigma(g) = hgh^{-1}</math>. The choice of <math>h</math> may depend on <math>g</math>.
 ==Relation with other properties==
@@ Line 23: / Line 11: @@
 ===Stronger properties===
-* [[Weaker than::Inner automorphism]]: {{proofofstrictimplicationat|[[Inner implies class-preserving]]|[[Class-preserving not implies inner]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::inner automorphism]] || conjugation by an element || [[inner implies class-preserving]] || [[class-preserving not implies inner]] || {{intermediate notions short|class-preserving automorphism|inner automorphism}}
+|-
+| [[Weaker than::locally inner automorphism]] || preserves conjugacy classes of finite tuples || [[locally inner implies class-preserving]] || [[class-preserving not implies locally inner]] || {{intermediate notions short|class-preserving automorphism|locally inner automorphism}}
+|}
 ===Weaker properties===
-* [[Stronger than::IA-automorphism]]: {{proofofstrictimplicationat|[[Class-preserving implies IA]]|[[IA not implies class-preserving]]}}
+{| class="sortable" border="1"
-* [[Stronger than::Normal automorphism]]: {{proofofstrictimplicationat|[[Class-preserving implies normal]]|[[Normal not implies class-preserving]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Center-fixing automorphism]]: {{proofofstrictimplicationat|[[Class-preserving implies center-fixing]]|[[Center-fixing not implies class-preserving]]}}
+|-
+| [[Stronger than::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]] || {{intermediate notions short|automorphism that preserves conjugacy classes for a generating set|class-preserving automorphism}}
+|-
+| [[Stronger than::IA-automorphism]] || induces identity map on [[abelianization]] || [[class-preserving implies IA]] || [[IA not implies class-preserving]] || {{intermediate notions short|IA-automorphism|class-preserving automorphism}}
+|-
+| [[Stronger than::normal automorphism]] || sends each normal subgroup to itself || [[class-preserving implies normal]] || [[normal not implies class-preserving]] || {{intermediate notions short|normal automorphism|class-preserving automorphism}}
+|-
+| [[Stronger than::weakly normal automorphism]] || sends each normal subgroup to a subgroup of itself || (via normal) || (via normal) || {{intermediate notions short|weakly normal automorphism|class-preserving automorphism}}
+|-
+| [[Stronger than::extended class-preserving automorphism]] || sends each element to conjugate or conjugate of inverse || || [[extended class-preserving not implies class-preserving]] || {{intermediate notions short|extended class-preserving automorphism|class-preserving automorphism}}
+|-
+| [[Stronger than::rational class-preserving automorphism]] || sends each element to conjugate of element generating same cyclic group || || || {{intermediate notions short|rational class-preserving automorphism|class-preserving automorphism}}
+|-
+| [[Stronger than::center-fixing automorphism]] || fixes every element of [[center]] || [[class-preserving implies center-fixing]] ||[[center-fixing not implies class-preserving]] || {{intermediate notions short|center-fixing automorphism|class-preserving automorphism}}
+|}
 ===Related properties===
 * [[Subgroup-conjugating automorphism]]: {{further|[[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==