Inner automorphisms are I-automorphisms in the variety of groups

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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 \sigma of a group G is termed an inner automorphism if there exists g \in G such that \sigma = c_g, where we define:

c_g(a) = gag^{-1}

I-automorphism

Further information: I-automorphism

Suppose \mathcal{V} is a variety of algebras, and A is an algebra in \mathcal{V}. An I-automorphism of A is an automorphism that can be expressed as:

x \mapsto \varphi(x,u_1,u_2,\dots,u_n)

where u_1, u_2, \dots, u_n \in A are fixed, and \varphi is a word in terms of the operations of the algebra,with the property that for any algebra B of \mathcal{V}, and any choice of values v_1,v_2,\dots,v_n \in B, the map:

x \mapsto \varphi(x,v_1,v_2,\dots,v_n)

gives an automorphism of B.

In other words \varphi is guaranteed to give an automorphism.

Facts used

Proof

Inner automorphisms are I-automorphisms

c_g(a) can be viewed as a word with input a and parameter g. This word gives an automorphism for every group and every choice of parameter g. Thus, inner automorphisms are I-automorphisms.

I-automorphisms are inner automorphisms

Given: A word \varphi(x,t_1,t_2,\dots,t_n) with the property that for any group G and any choice of values of u_i in G, the map sending x to \varphi(x,u_1,u_2,\dots,u_n) is an automorphism

To prove: This automorphism is always inner.

Proof: Let F be the free group on n+2 generators, and let u_1,u_2,\dots,u_n,u_{n+1},u_{n+2} be a freely generating set for F. By the given condition, the map:

x \mapsto \varphi(x,u_1,u_2,\dots,u_n)

gives an automorphism of F. In particular:

\varphi(u_{n+1},u_1,u_2,\dots,u_n)\varphi(u_{n+2},u_1,u_2,\dots,u_n) = \varphi(u_{n+1}u_{n+2},u_1,u_2,\dots,u_n)

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A little manipulation of possible expressions shows that \varphi must be of the form:

\varphi(x,u_1,u_2,\dots,u_n) \equiv \psi(u_1,u_2,\dots,u_n)x\psi(u_1,u_2,\dots,u_n)^{-1}

Hence, any automorphism obtained using \varphi must be an inner automorphism.