Inner automorphism

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 \[(f \circ \text{Int}(a) \circ f^{-1})(x) = f\bigl( a f^{-1}(x)a^{-1} \bigr) = f(a)x \bigl[ f(a) \bigr]^{-1} = \text{Int}\bigl(f(a)\bigr) (x) .\] Thus $\text{Int}(G)$ is a normal subgroup of $\text{Aut}(G)$. $\blacksquare$

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