groups of order pq

We can use Sylow’s theorems to examine a group G of order pq, where p and q are primes ( and p<q.

Let np and nq denote, respectively, the number of Sylow p-subgroupsMathworldPlanetmathPlanetmath and Sylow q-subgroups of G.

Sylow’s theorems tell us that nq=1+kq for some integer k and nq divides pq. But p and q are prime and p<q, so this implies that nq=1. So there is exactly one Sylow q-subgroup, which is therefore normal (indeed, fully invariant) in G.

Denoting the Sylow q-subgroup by Q, and letting P be a Sylow p-subgroup, then QP={1} and QP=G, so G is a semidirect productMathworldPlanetmath of Q and P. In particular, if there is only one Sylow p-subgroup, then G is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of Q and P, and is therefore cyclic.

Given G=QP, it remains to determine the action of P on Q by conjugationMathworldPlanetmath. There are two cases:

Case 1: If p does not divide q-1, then since np=1+mp cannot equal q we must have np=1, and so P is a normal subgroupMathworldPlanetmath of G. This gives G=Cp×Cq a direct product, which is isomorphicPlanetmathPlanetmathPlanetmath to the cyclic groupMathworldPlanetmath Cpq.

Case 2: If p divides q-1, then Aut(Q)Cq-1 has a unique subgroup ( P of order p, where P={xxii/q,ip=1}. Let a and b be generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for P and Q respectively, and suppose the action of a on Q by conjugation is xxi0, where i01 in /q. Then G=a,bap=bq=1,aba-1=bi0. Choosing a different i0 amounts to choosing a different generator a for P, and hence does not result in a new isomorphism class. So there are exactly two isomorphism classes of groups of order pq.

Title groups of order pq
Canonical name GroupsOfOrderPq
Date of creation 2013-03-22 12:51:05
Last modified on 2013-03-22 12:51:05
Owner yark (2760)
Last modified by yark (2760)
Numerical id 22
Author yark (2760)
Entry type Example
Classification msc 20D20
Related topic SylowTheorems
Related topic SemidirectProductOfGroups