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

$c g (a) = g a g - 1$

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: