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Inner automorphism

From AoPSWiki

An inner automorphism is an automorphism on a group $G$ of the form $x \mapsto axa^{-1}$ , for some $a$ in $G$ . This mapping is denoted $\text{Int}(a)$ . Every such mapping is an automorphism.

Sometimes $\text{Int}(a)(x)$ is denoted as $^ax$ , or as $x^{a^{-1}}$ .

Theorem. For every $a$ in $G$ , $\text{Int}(a)$ is a group automorphism on $G$ . Furthermore, the mapping $\text{Int}:a \mapsto \text{Int}(a)$ is a group homomorphism from $G$ to $\text{Aut}(G)$ , the group of automorphisms on $G$ . Its kernel is the center of $G$ , and its image, the set of inner automorphisms, is a normal subgroup of $\text{Aut}(G)$ .

Proof. Let $a$ be an element of $G$ . Since $G$ is a group, $axa^{-1} = aya^{-1}$ if and only if $x=y$ , so $\text{Int}(a)$ is injective. Every element $x$ has an inverse image $a^{-1}xa$ as well, so $\text{Int}(a)$ is surjective onto $G$ . Finally, for all $x,y \in G$ , $\text{Int}(a)(xy) = axya^{-1} = (axa^{-1})(aya^{-1}) = \text{Int}(a)(x) \text{Int}(a)(y),$ so $\text{Int}(a)$ is an endomorphism of $G$ . Therefore it is an automorphism of $G$ .

Since $\bigl[ \text{Int}(x) \circ \text{Int}(y) \bigr](z) = xyzy^{-1}x^{-1} = (xy)z(xy)^{-1} = \text{Int}(xy)(z),$ $\text{Int}$ is a group homomorphism from $G$ to $\text{Aut}(G)$ . Note that $\text{Int}(a)$ is the identity map on $G$ if and only if, for all $x\in G$ , $axa^{-1} = x = xaa^{-1}$ , which is true if and only if $ax = xa$ , which is true if and only if $a$ is in the center of $G$ .

Finally, if $f$ is any automorphism of $G$ , $\text{Int}(a)$ is an inner automorphism on $G$ , and $x$ is any element of $G$ , then $(f \circ \text{Int}(a) \circ f^{-1})(x) = f\bigl( a f^{-1}(x)a^{-1} \bigr) = f(a)x \bigl[ f(a) \bigr]^{-1} = \text{Int}\bigl(...$ Thus $\text{Int}(G)$ is a normal subgroup of $\text{Aut}(G)$ . $\blacksquare$

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