Inner implies class-preserving: Difference between revisions
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{{trivial result}} | {{trivial result}} | ||
{{automorphism property implication}} | {{automorphism property implication| | ||
stronger = inner automorphism| | |||
weaker = class-preserving automorphism}} | |||
==Statement== | ==Statement== | ||
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===Property-theoretic statement=== | ===Property-theoretic statement=== | ||
The [[automorphism property]] of being an [[inner automorphism]] is ''stronger than'', or ''implies'', the automorphism property of being a [[class automorphism]]. | The [[automorphism property]] of being an [[inner automorphism]] is ''stronger than'', or ''implies'', the automorphism property of being a [[class-preserving automorphism]] (also called class automorphism). | ||
===Verbal statement=== | ===Verbal statement=== | ||
Any inner automorphism of a group is a class automorphism: it sends every element to its conjugacy class. | Any inner automorphism of a group is a class automorphism: it sends every element to its [[conjugacy class]]. | ||
==Definitions used== | ==Definitions used== | ||
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An automorphism <math>\sigma</math> of a group <math>G</math> is termed an inner automorphism if there exists <math>h \in G</math> such that for every <math>g \in G</math>, <math>\sigma(g) = hgh^{-1}</math> | An automorphism <math>\sigma</math> of a group <math>G</math> is termed an inner automorphism if there exists <math>h \in G</math> such that for every <math>g \in G</math>, <math>\sigma(g) = hgh^{-1}</math> | ||
===Class automorphism=== | ===Class-preserving automorphism=== | ||
{{further|[[Class automorphism]]}} | {{further|[[Class-preserving automorphism]]}} | ||
An automorphism <math>\sigma</math> of a group <math>G</math> is termed a class automorphism if, for every <math>g \in G</math>, there exists <math>h \in G</math> such that <math>\sigma(g) = hgh^{-1}</math> | An automorphism <math>\sigma</math> of a group <math>G</math> is termed a class automorphism if, for every <math>g \in G</math>, there exists <math>h \in G</math> such that <math>\sigma(g) = hgh^{-1}</math> | ||
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The converse of this statement is not true. | The converse of this statement is not true. | ||
{{further|[[Class not implies inner]]}} | {{further|[[Class-preserving not implies inner]]}} | ||
==Proof== | ==Proof== | ||
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* [[Center-fixing automorphism]]: This fixes every element in the center of the group | * [[Center-fixing automorphism]]: This fixes every element in the center of the group | ||
* [[IA-automorphism]]: This acts as the identity on the [[Abelianization]] of the group | * [[IA-automorphism]]: This acts as the identity on the [[Abelianization]] of the group | ||
* [[ | * [[Normal automorphism]]: This is an automorphism that preserves every [[normal subgroup]] | ||
Latest revision as of 20:26, 9 August 2008
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This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., inner automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)
View all automorphism property implications | View all automorphism property non-implications
Get more facts about inner automorphism|Get more facts about class-preserving automorphism
Statement
Property-theoretic statement
The automorphism property of being an inner automorphism is stronger than, or implies, the automorphism property of being a class-preserving automorphism (also called class automorphism).
Verbal statement
Any inner automorphism of a group is a class automorphism: it sends every element to its conjugacy class.
Definitions used
Inner automorphism
Further information: Inner automorphism
An automorphism of a group is termed an inner automorphism if there exists such that for every ,
Class-preserving automorphism
Further information: Class-preserving automorphism
An automorphism of a group is termed a class automorphism if, for every , there exists such that
Converse
The converse of this statement is not true. Further information: Class-preserving not implies inner
Proof
Hands-on proof
Given: A group , an inner automorphism of , an element
To prove: There exists such that
Proof: In fact, by the definition of inner automorphism, we do have a , that doesn't even depend on the choice of .
Deeper insight into the proof
One way of viewing the condition of being a class automorphism is: it looks like an inner automorphism locally at every element. In other words, if we're looking at just one element at a time, the automorphism looks like an inner automorphism. The problem is that the choice of conjugating element may differ depending on which element of the group we're looking at.
Related properties
Other related properties, all of which are weaker than the property of being a class automorphism:
- Subgroup-conjugating automorphism: This sends every subgroup to a conjugate subgroup
- Center-fixing automorphism: This fixes every element in the center of the group
- IA-automorphism: This acts as the identity on the Abelianization of the group
- Normal automorphism: This is an automorphism that preserves every normal subgroup