Group Automorphisms Review

# Group Automorphisms Review

We will now review some of the recent material regarding group automorphisms.

• On the Group Automorphisms page we said that if $(G, *)$ is a group, then a group Automorphism is a function $f : G \to G$ that has the following properties:
• (a) $f$ is a bijection.
• (b) For all $x, y \in G$ we have that:
(1)
\begin{align} \quad f(x * y) = f(x) * f(y) \end{align}
• In other words, a group automorphism is a group isomorphism from a group to itself.
(2)
\begin{align} \quad \mathrm{Aut} (G) = \{ f : f \: \mathrm{is \: an \: automorphism \: of \:} G \} \end{align}
• We proved that the set $\mathrm{Aut} (G)$ with the operation of function composition $\circ$ is itself a group.