Inner automorphisms are I-automorphisms in the variety of groups
Contents
Statement
In the Variety of groups (?), treated as a variety of algebras, an I-automorphism (?) is the same thing as an Inner automorphism (?).
Definitions used
Inner automorphism
Further information: Inner automorphism
An automorphism of a group
is termed an inner automorphism if there exists
such that
, where we define:
I-automorphism
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.
Facts used
- Inner automorphisms actually are automorphisms. For full proof, refer: Group acts as automorphisms by conjugation
Proof
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:
A little manipulation of possible expressions shows that must be of the form:
Hence, any automorphism obtained using must be an inner automorphism.