Classification of groups of order a product of a prime-square and another prime

From Groupprops
Jump to: navigation, search

Statement

Suppose p and q are distinct prime numbers. This article classifies the groups of order p^2q.

Case that p does not divide q - 1 and q does not divide p - 1

In this case, p^2q is an abelianness-forcing number, i.e., all groups of this order are abelian. The two abelian groups are:

Group GAP ID second part Abelian? p-Sylow subgroup Is the p-Sylow subgroup normal? Is the q-Sylow subgroup normal?
cyclic group of order p^2q 1 Yes cyclic group of prime-square order Yes Yes
direct product of cyclic group of order p and cyclic group of order pq, also same as direct product of elementary abelian group of order p^2 and cyclic group of order q 2 Yes elementary abelian group of prime-square order Yes Yes

Case that p divides q - 1 but p^2 does not divide q - 1 and q does not divide p^2 - 1

In this case, there are four isomorphism classes of groups of order p^2q, given as follows:

Group GAP ID second part Abelian? p-Sylow subgroup Is the p-Sylow subgroup normal? Is the q-Sylow subgroup normal?
cyclic group of order p^2q 1 Yes cyclic group of prime-square order Yes Yes
semidirect product of cyclic group of order q by cyclic group of order p^2, where the action by conjugation of a generator is an automorphism of order p 2 No cyclic group of prime-square order No Yes
semidirect product of cyclic group of order pq by cyclic group of order p, where the action by conjugation of a generator is an automorphism of order p 3 No elementary abelian group of prime-square order No Yes
direct product of cyclic group of order p and cyclic group of order pq, also same as direct product of elementary abelian group of order p^2 and cyclic group of order q 4 Yes elementary abelian group of prime-square order Yes Yes

Case that p^2 divides q - 1

In this case, there are five isomorphism classes of groups of order p^2q, given as follows:

Group GAP ID second part Abelian? p-Sylow subgroup Is the p-Sylow subgroup normal? Is the q-Sylow subgroup normal?
cyclic group of order p^2q 1 Yes cyclic group of prime-square order Yes Yes
semidirect product of cyclic group of order q by cyclic group of order p^2, where the action by conjugation of a generator is an automorphism of order p 2 No cyclic group of prime-square order No Yes
semidirect product of cyclic group of order q by cyclic group of order p^2, where the action by conjugation of a generator is an automorphism of order p^2 3 No cyclic group of prime-square order No Yes
semidirect product of cyclic group of order pq by cyclic group of order p, where the action by conjugation of a generator is an automorphism of order p 4 No elementary abelian group of prime-square order No Yes
direct product of cyclic group of order p and cyclic group of order pq, also same as direct product of elementary abelian group of order p^2 and cyclic group of order q 5 Yes elementary abelian group of prime-square order Yes Yes

Case that q divides p - 1 but does not divide p + 1

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Case that q divides p + 1 but does not divide p - 1

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Special cases: order 12 and order 18

There are five groups of order 12. See classification of groups of order 12. The unusual example is alternating group:A4.

There are five groups of order 18. See classification of groups of order 18.