# example of groups of order pq

As a specific example, let us classify groups of order 21. Let $G$ be a group of order 21. There is only one Sylow 7-subgroup $K$ so it is normal. The possibility of there being conjugate Sylow 3-subgroups is not ruled out. Let $x$ denote a generator^{} for $K$, and $y$ a generator for one of the Sylow 3-subgroups $H$. Then ${x}^{7}={y}^{3}=1$, and $yx{y}^{-1}={x}^{i}$ for some $$ since $K$ is normal. Now $x={y}^{3}x{y}^{-3}={y}^{2}{x}^{i}{y}^{-2}=y{x}^{{i}^{2}}{y}^{-1}={x}^{{i}^{3}}$, or ${i}^{3}=1mod7$. This implies $i=1,2$, or 4.

Case 1: $yx{y}^{-1}=x$, so $G$ is abelian^{} and isomorphic^{} to
${C}_{21}={C}_{3}\times {C}_{7}$.

Case 2: $yx{y}^{-1}={x}^{2}$, then every product of the elements $x,y$ can be reduced to one in the form ${x}^{i}{y}^{j}$, $$, $$. These 21 elements are clearly distinct, so $G=\u27e8x,y\mid {x}^{7}={y}^{3}=1,yx={x}^{2}y\u27e9$.

Case 3: $yx{y}^{-1}={x}^{4}$, then since ${y}^{2}$ is also a generator of $H$ and ${y}^{2}x{y}^{-2}=y{x}^{4}{y}^{-1}={x}^{16}={x}^{2}$, we have recovered case 2 above.

Title | example of groups of order pq |
---|---|

Canonical name | ExampleOfGroupsOfOrderPq |

Date of creation | 2013-03-22 14:51:15 |

Last modified on | 2013-03-22 14:51:15 |

Owner | jh (7326) |

Last modified by | jh (7326) |

Numerical id | 4 |

Author | jh (7326) |

Entry type | Example |

Classification | msc 20D20 |