# If $p <q$ (primes), how to classify the semi-direct products of $\mathbb{Z}_{q}$ by $\mathbb{Z}_{p}$?

I have solved several exercises on classifying groups and I have been wanting to generalize my results. however, I came across this problem where I know that there is no semi-direct products of $$\mathbb{Z}_{p}$$ by $$\mathbb{Z}_{q}$$, only direct products, since $$p, but, I had difficulty classifying the semi-direct products of $$\mathbb{Z}_{q}$$ by $$\mathbb{Z}_{p}$$.

• Just to be clear, you are looking for $\Bbb Z_q\rtimes \Bbb Z_p$? – Alex Ortiz Nov 5 '18 at 4:49
• @AOrtiz exactly – Erick Vinicius Nov 5 '18 at 5:04

We need to find all the homomorphisms $$\Bbb Z_p\to \mathrm{Aut}(\Bbb Z_q)$$. Recall that $$\mathrm{Aut}(\Bbb Z_q) \cong \Bbb Z_{q-1}$$ since $$q$$ is prime. Hence we need to know all the homomorphisms $$\phi\colon \Bbb Z_p\to\Bbb Z_{q-1}$$, each of which is determined by where we send the generator $$1$$ of $$\Bbb Z_p$$. For any such homomorphism, the order of $$\phi(1)$$ divides $$p$$ and it divides $$q-1$$. One option for $$\phi$$ is the trivial homomorphism, which corresponds to $$1\mapsto \mathrm{Id}_{\Bbb Z_q}$$, and that gives us the usual direct product $$\Bbb Z_q\times\Bbb Z_p$$.
The other option is $$p$$ divides $$q-1$$, i.e. where $$q \equiv 1\bmod p$$. Then the image $$\phi(1)$$ generates a unique cyclic subgroup of $$\Bbb Z_{q-1}$$ of order $$p$$, generated by $$\frac{q-1}{p}$$. This corresponds to the automorphism $$\psi$$ of $$\Bbb Z_q$$ defined by $$\psi\colon 1\mapsto \frac{q-1}{p}$$.
Hence the isomorphism classes of groups $$G$$ of order $$pq$$, with $$p < q$$ and $$p,q$$ primes are $$G\cong \Bbb Z_q\times\Bbb Z_p\qquad\text{or}\qquad G\cong \Bbb Z_q\rtimes_\phi\Bbb Z_p,$$ where $$\phi\colon \Bbb Z_p\to \mathrm{Aut}(\Bbb Z_q)$$ is defined by $$\phi\colon 1 \mapsto \psi$$, where $$\psi$$ is the automorphism of $$\Bbb Z_q$$ we described above.