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

From Groupprops

## Contents

## Statement

Suppose and are *distinct* prime numbers. This article classifies the groups of order .

### Case that does not divide and does not divide

In this case, 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? | -Sylow subgroup | Is the -Sylow subgroup normal? | Is the -Sylow subgroup normal? |
---|---|---|---|---|---|

cyclic group of order | 1 | Yes | cyclic group of prime-square order | Yes | Yes |

direct product of cyclic group of order and cyclic group of order , also same as direct product of elementary abelian group of order and cyclic group of order | 2 | Yes | elementary abelian group of prime-square order | Yes | Yes |

### Case that divides but does not divide and does not divide

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

Group | GAP ID second part | Abelian? | -Sylow subgroup | Is the -Sylow subgroup normal? | Is the -Sylow subgroup normal? |
---|---|---|---|---|---|

cyclic group of order | 1 | Yes | cyclic group of prime-square order | Yes | Yes |

semidirect product of cyclic group of order by cyclic group of order , where the action by conjugation of a generator is an automorphism of order | 2 | No | cyclic group of prime-square order | No | Yes |

semidirect product of cyclic group of order by cyclic group of order , where the action by conjugation of a generator is an automorphism of order | 3 | No | elementary abelian group of prime-square order | No | Yes |

direct product of cyclic group of order and cyclic group of order , also same as direct product of elementary abelian group of order and cyclic group of order | 4 | Yes | elementary abelian group of prime-square order | Yes | Yes |

### Case that divides

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

Group | GAP ID second part | Abelian? | -Sylow subgroup | Is the -Sylow subgroup normal? | Is the -Sylow subgroup normal? |
---|---|---|---|---|---|

cyclic group of order | 1 | Yes | cyclic group of prime-square order | Yes | Yes |

semidirect product of cyclic group of order by cyclic group of order , where the action by conjugation of a generator is an automorphism of order | 2 | No | cyclic group of prime-square order | No | Yes |

3 | No | cyclic group of prime-square order | No | Yes | |

4 | No | elementary abelian group of prime-square order | No | Yes | |

direct product of cyclic group of order and cyclic group of order , also same as direct product of elementary abelian group of order and cyclic group of order | 5 | Yes | elementary abelian group of prime-square order | Yes | Yes |

### Case that divides but does not divide

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### Case that divides but does not divide

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### 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.