Not every group of order $pq$ (where $p$ and $q$ are distinct primes) is necessarily cyclic, but under certain conditions, such groups are guaranteed to be cyclic. Let us analyze this carefully:

Lagrange's Theorem and Basic Setup:

1. Let $G$ be a group of order $pq$, where $p$ and $q$ are distinct primes such that $p < q$.
2. By Lagrange's theorem, the order of any subgroup of $G$ must divide $pq$, so the possible orders of subgroups are $1$, $p$, $q$, and $pq$.

The Key Idea:

A group of order $pq$ is cyclic if and only if one of the following conditions holds:

• $p \nmid (q-1)$, i.e., $p$ does not divide $q-1$.

Reason:

1. If $p \nmid (q-1)$, the Sylow $q$-subgroup is unique. This forces $G$ to be cyclic.
2. If $p \mid (q-1)$, there may be multiple Sylow subgroups, and $G$ may not be cyclic. In this case, $G$ could be a non-cyclic group like a semidirect product.

Outline of Proof:

Case 1: Cyclic

If $G$ is cyclic, it is generated by a single element, and all elements are powers of this generator. For groups of order $pq$, a cyclic structure arises naturally under the divisibility conditions.

Case 2: Non-cyclic

When $p \mid (q-1)$, $G$ can have a more complex structure, potentially forming a non-abelian group (e.g., the dihedral group).

Conclusion:

To summarize, a group of order $pq$ is cyclic if and only if $p \nmid (q-1)$. If $p \mid (q-1)$, the group may still be cyclic, but it can also have other structures.

Would you like to delve deeper into the proofs, examples, or the structure of groups of order $pq$?

Further Questions:

1. Can you show an example of a cyclic group of order $pq$?
2. What happens when $p \mid (q-1)$?
3. How do Sylow theorems help in proving these results?
4. Can we generalize this to groups of order $p^nq$?
5. What are the implications for groups where $q < p$?

Tip:

Always verify divisibility conditions like $p \mid (q-1)$ when analyzing group cyclicity for composite orders.