Consider the group $\mbox{Aff}(p)$ of automorphisms of $\mathbb{A}^1(p)$ [that is, the transformations $x\rightarrow a x + b,$ for $a\neq 0 \mod p.$] Is it true that a random pair of elements generate $\mbox{Aff}(p)?,$ for large $p?$



Is the affine group generically 2-generated?
In general not. Consider the homomorphism $\mathrm{Aff}(p)\rightarrow\mathbb{F}\_p^\*$ mapping $ax+b$ to $a$. The probability that two random elements generate a subgroup for which the induced map is still surjective equals the probability that two random elements in $\mathbb{F}\_p^\*$ generate this group, which is $$\frac{1}{(p-1)^2}\sum\_{d|p-1} \mu\left(\frac{p-1}{d}\right) d^2 = \sum\_{t|p-1}\frac{\mu(t)}{t^2} = \prod\_{q|p-1}\left(1-\frac{1}{q^2}\right),$$
where in the last product $q$ runs over primes only. Depending on the prime factors of $p-1$ this quantity is somewhere in the interval $[\frac{6}{\pi^2}, \frac{3}{4}]$.


On the other hand two random elements almost surely generate a subgroup containing a translation, and therefore all translations, thus the probability that two random elements generate $\mathrm{Aff}(p)$ is asymptotically equal to the quantity given above.


