Is the following statement true? A group of order pq (p < q) is always cyclic

No. S₃ of order 6 = 2·3 Non-abelian because 2 | 3 − 1. The cyclic conclusion needs p ∤ q − 1.

Counterexample

Is this true?

A group of order pq (p < q) is always cyclic

No — it is false.

The counterexample

$S_{3}$ of order 6 = 2·3

Why it fails

Non-abelian because 2 | 3 − 1. The cyclic conclusion needs p ∤ q − 1.

Keep practicing

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.