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 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''.
-===Origin of the term===
-The term ''class automorphism'' was used in the ''Journal of Algebra'' in some papers on class automorphisms.
 ==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 [[automorphism]] of a group is termed a '''class automorphism''' if it takes each element to within its [[conjugacy class]].
-===Definition with symbols===
-An automorphism <math>\sigma</math> of a group <math>G</math> is termed a '''class 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 27: / Line 11: @@
 ===Stronger properties===
-* [[Inner automorphism]]
+{| 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===
-* [[IA-automorphism]]
+{| class="sortable" border="1"
-* [[Center-fixing automorphism]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[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]]
+* [[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==
@@ Line 42: / Line 52: @@
 {{group-closed}}
-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 automorphism]]s.
+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 automorphism]]s, and is a [[normal subgroup]] inside the automorphism group.
-{{normal subgroup of aut-group}}
-The group of class automorphisms forms a normal subgroup of the automorphism group. This can be proved from first principles, although it also follows from the more general fact that for any [[group-closed automorphism property]], the group of automorphisms satisfying it is a normal subgroup of the automorphism group.
 {{dirprodclosed ap}}
@@ Line 53: / Line 59: @@
 Here, <math>\sigma_1 \times \sigma_2</math> is the automorphism of <math>G_1 \times G_2</math> that acts as <math>\sigma_1</math> on the first coordinate and <math>\sigma_2</math> on the second.
+==References==
+* {{paperlink|Burnside13}}
+* {{paperlink|Wall47}}
+* {{paperlink|Hertweck01}}
+* {{paperlink|Yadav07}}
+==External links==
+{{searchbox|"class+preserving+automorphism"}}