Inner automorphisms are I-automorphisms in the variety of groups

Statement
In the fact about::variety of groups, treated as a variety of algebras, an fact about::I-automorphism is the same thing as an fact about::inner automorphism.

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
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

 * Inner automorphisms actually are automorphisms.

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:

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

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.