Frobenius group (reviewed)

Given a prime power $q$, the Frobenius group $F_q$ is the group of invertible affine transformations ($x \mapsto ax + b$) on $\F_q$. Explicitly, it is the group $$ F_q := \{ \varphi : \F_q \to \F_q \mid \exists a, b \in \F_q \text{ such that } a \neq 0 \text{ and } \varphi(x) = ax + b \text{ for all } x \in \F_q \} $$ where the usual composition of maps is the group operation. It has order $q(q-1)$ and is a Z-group (and hence is metacyclic and solvable).