How to show a group G with a given presentation is the semidirect product of two cyclic groups of prime order?

Question

The question as posed in my homework assignment is as follows. I have been able to prove that the map $\sigma$ is an automorphism, and I think that the existence of the integer $t$ must be related to Fermat's Little Theorem, but after that I'm completely lost. Should I be demonstrating that $C_p \rtimes_{\phi} C_q$ has the same presentation as $G$?

Exercise: Let $G$ and $H$ be groups and let $\phi : H \to \text{Aut } (G)$ be a homomorphism. We define the semi-direct product of $G$ and $H$ (with respect to the homomorphism $\phi$) to be the cartesian product of $G$ and $H$ together with the following operation: \begin{align*} (g_1,h_1)(g_2,h_2) = (g_1[\phi(h_1)](g_2),h_1h_2). \end{align*} The semi-direct product of $G$ and $H$ with respect to $\phi$ is denoted $G \ltimes_{\phi} H$.

Let $p$ and $q$ be distinct primes with $p > q$ and $p \equiv 1 \mod q$. Let $G$ be the group determined by the generators $a$ and $b$ subject to the relations $|a| = p$, $|b| = q$, and $ba = a^t b$, where $t$ is a fixed integer satisfying $t^q \equiv 1 \mod p$, $t \not\equiv 1 \mod p$ (such an integer must exist - why?). Show that $G$ is a semi-direct product of the cyclic groups $C_p = \langle a \rangle$ and $C_q = \langle b \rangle$. [Hint: first show that the map $\sigma : C_p \to C_p$ given by $\sigma (a^j) = a^{jt}$ is an automorphism, and then consider the map $\phi : C_q \to \text{Aut } (C_p)$ given by $\phi (b^i) = \sigma ^i$.] The group $G$ is called a \textit{metacyclic} group.

Answer 1

Comparing presentations is a very bad way of proving that two groups are isomorphic. Indeed, groups can have many different presentations. (In fact, the problem of determining whether a given presentation gives the trivial group is already very difficult.)

If you want to prove that two groups are isomorphic, you should first obtain a morphism from one to the other. This will be your "candidate". Then, you want to prove that your candidate is an isomorphism (if it is not, find a new candidate).

For instance, you could take the following map as a candidate. Let $u, v$ be generators of $C_p$ and $C_q$ respectively. Define

$$f: C_p \rtimes C_q \to G$$ by $$(u^s, v^t) \mapsto a^sb^t.$$

In order, you should check the following: Is it a well-defined map? Is it a group homomorphism? Is it injective? Is it surjective? If you have answered yes to all of these questions, then you are done. Otherwise, you might want to tinker around with the definition until you find something that works.