Inner implies class-preserving
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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)
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Any inner automorphism of a group is a class automorphism: it sends every element to its conjugacy class.
Further information: Inner automorphism
An automorphism of a group is termed an inner automorphism if there exists such that for every ,
Further information: Class-preserving automorphism
An automorphism of a group is termed a class automorphism if, for every , there exists such that
The converse of this statement is not true. Further information: Class-preserving not implies inner
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.
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