Inner automorphisms are I-automorphisms in the variety of groups
Further information: Inner automorphism
An automorphism of a group is termed an inner automorphism if there exists such that , where we define:
Further information: I-automorphism
Suppose is a variety of algebras, and is an algebra in . An I-automorphism of is an automorphism that can be expressed as:
where are fixed, and is a word in terms of the operations of the algebra,with the property that for any algebra of , and any choice of values , the map:
gives an automorphism of .
In other words is guaranteed to give an automorphism.
- Inner automorphisms actually are automorphisms. For full proof, refer: Group acts as automorphisms by conjugation
Inner automorphisms are I-automorphisms
can be viewed as a word with input and parameter . This word gives an automorphism for every group and every choice of parameter . Thus, inner automorphisms are I-automorphisms.
I-automorphisms are inner automorphisms
Given: A word with the property that for any group and any choice of values of in , the map sending to is an automorphism
To prove: This automorphism is always inner.
Proof: Let be the free group on generators, and let be a freely generating set for . By the given condition, the map:
gives an automorphism of . In particular:
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A little manipulation of possible expressions shows that must be of the form:
Hence, any automorphism obtained using must be an inner automorphism.