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 simply an isomorphism from $G$ onto $G$, that is, for all $x, y \in G$ we have that:

(1)

\begin{align} \quad f(x * y) = f(x) * f(y) \end{align}

• On The Group of Automorphisms of a Group, Aut(G) we defined the set of all automorphisms on $G$ to be denoted by $\mathrm{Aut}(G)$ and we proved that it forms a group with the operation of composition.

• On the Inner Automorphisms of a Group page we defined an automorphism of $G$ to be an inner automorphism of $G$ if it is of the form $i_a$ with $a \in G$, where $i_a : G \to G$ is defined for all $g \in G$ by:

(2)

\begin{align} \quad i_a(g) = aga^{-1} \end{align}

• On The Subgroup of Inner Automorphisms of a Group, Inn(G) page we defined the group of all inner automorphisms to be denoted by $\mathrm{Inn}(G)$, and we proved that it is a subgroup of $\mathrm{Aut}(G)$.

• We proved that the set $\mathrm{Aut} (G)$ with the operation of function composition $\circ$ is itself a group.