# Difference between revisions of "Classification of groups of order four times a prime congruent to 3 modulo 4"

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==Statement== | ==Statement== | ||

− | Suppose <math>p</math> is an odd prime that is congruent to 3 modulo 4, i.e., 4 divides <math>p - 3</math>. Suppose further that <math>p > 3</math> | + | Suppose <math>p</math> is an odd prime that is congruent to 3 modulo 4, i.e., 4 divides <math>p - 3</math>. Suppose further that <math>p > 3</math>. |

Then, there are four isomorphism classes of groups of order <math>4p</math>, as detailed below: | Then, there are four isomorphism classes of groups of order <math>4p</math>, as detailed below: | ||

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| direct product of cyclic group of order <math>2p</math> and [[cyclic group:Z2]] (also, direct product of [[group of prime order]] and [[Klein four-group]] || 4 || Yes || [[Klein four-group]] || Yes || Yes | | direct product of cyclic group of order <math>2p</math> and [[cyclic group:Z2]] (also, direct product of [[group of prime order]] and [[Klein four-group]] || 4 || Yes || [[Klein four-group]] || Yes || Yes | ||

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+ | The case <math>p=3</math> differs from tthe general case. For that case, see [[classification of group of order 12]]. | ||

==Related classifications== | ==Related classifications== |

## Latest revision as of 21:21, 15 June 2012

## Statement

Suppose is an odd prime that is congruent to 3 modulo 4, i.e., 4 divides . Suppose further that .

Then, there are four isomorphism classes of groups of order , as detailed below:

Group | Second part of GAP ID | Abelian? | Isomorphism class of 2-Sylow subgroup | Is the 2-Sylow subgroup normal? | Is the -Sylow subgroup normal? |
---|---|---|---|---|---|

dicyclic group of order | 1 | No | cyclic group:Z4 | No | Yes |

cyclic group of order | 2 | Yes | cyclic group:Z4 | Yes | Yes |

dihedral group of order | 3 | No | Klein four-group | No | Yes |

direct product of cyclic group of order and cyclic group:Z2 (also, direct product of group of prime order and Klein four-group | 4 | Yes | Klein four-group | Yes | Yes |

The case differs from tthe general case. For that case, see classification of group of order 12.