View source for Inner implies class-preserving

{{trivial result}}
{{automorphism property implication|
stronger = inner automorphism|
weaker = 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|[[Inner automorphism]]}}

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-preserving 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>

==Converse==

The converse of this statement is not true.
{{further|[[Class-preserving not implies inner]]}}

==Proof==

===Hands-on proof===

''Given'': A group <math>G</math>, an inner automorphism <math>\sigma</math> of <math>G</math>, an element <math>g \in G</math>

''To prove'': There exists <math>h \in G</math> such that <math>\sigma(g) = hgh^{-1}</math>

''Proof'': In fact, by the definition of inner automorphism, we do have a <math>h</math>, that doesn't even depend on the choice of <math>g</math>.

===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]]