Groups of Order pq

Abstract Algebra · Axiom Academy

Complete classification of groups whose order is the product of two distinct primes The answer depends entirely on whether q is congruent to 1 modulo p: If q ≢ 1 (mod p), there is exactly one group of order pq (up to isomorphism), namely ℤ pq ≅ ℤ p × ℤ q If q ≡ 1 (mod p), there are exactly two non-isomorphic groups: the cyclic group ℤ pq and one non-abelian group The classification relies on the Sylow theorems. For a group G of order pq: When does n q = 1? This is equivalent to asking: when is there no way to have p distinct q-Sylow subgroups? From Sylow's theorem, if n q = p, then p ≡ 1 (mod q). But since p q = p is only possible when we can conjugate the q-Sylow subgroups, which requires non-trivial automorphisms. 4. Constructing the Non-Abelian Group When q ≡ 1 (mod p), we construct the non-abelian group as a semidirect product: The key relation in this group is: aba -1 = b r , which makes the group non-abelian. Let's examine specific cases to see the theorem in action:

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