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:

\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.

- On
**The Group of Automorphisms of a Group**we defined the set of all group automorphisms on a group $(G, *)$ to be:

\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.