Inner automorphisms are I-automorphisms in the variety of groups

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

Inner automorphisms actually are automorphisms. For full proof, refer: Group acts as automorphisms by conjugation

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.