Inner Automorphisms of a Group

Inner Automorphisms of a Group

Recall from The Group of Automorphisms of a Group, Aut(G) page that if $G$ is a group then the set of all isomorphisms from $G$ to $G$ (called Automorphisms) with the operation forms a group denoted $\mathrm{Aut}(G)$ called the automorphism group of $G$.

We now classify a special type of automorphism on $G$ called an inner automorphism.

Definition: Let $G$ be a group. An Inner Automorphism on $G$ is an automorphism of $G$ of the form $i_a$ for some $a \in G$, where $i_a : G \to G$ is defined for all $g \in G$ by $i_a(g) = aga^{-1}$.

The following proposition tells us that each $i_a$ is indeed an automorphism of $G$.

Proposition 1: Let $G$ be a group. Then for each $a \in G$, $i_a : G \to G$ defined for all $g \in G$ by $i_a(g) = aga^{-1}$ is an automorphism of $G$.
  • Proof: Fix $a \in G$. Observe first that $i_a : G \to G$ is indeed a homomorphism since for all $g_1, g_2 \in G$ we have that:
\begin{align} \quad i_a(g_1g_2) = ag_1g_2a^{-1} = ag_1(a^{-1}a)g_2a^{-1} = (ag_1a^{-1})(ag_2a^{-1}) = i_a(g_1) i_a(g_2) \end{align}
  • Now observe that $i_a$ is injective, since if $g_1, g_2 \in G$ is such that $i_a(g_1) = i_a(g_2)$ then $ag_1a^{-1} = ag_2a^{-1}$ which implies that $g_1 = g_2$.
  • Lastly, observe that $i_a$ is surjective since if $g \in G$ then $a^{-1}ga \in G$ is such that $i_a(a^{-1}ga) = a(a^{-1}ga)a^{-1} = g$.
  • Thus $i_a$ is a bijective homomorphism of $G$ to $G$, i.e., $i_a$ is an automorphism of $G$. $\blacksquare$
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