# groups of order pq

We can use Sylow’s theorems to examine a group $G$ of order $pq$, where $p$ and $q$ are primes (http://planetmath.org/Prime) and $$.

Let ${n}_{p}$ and ${n}_{q}$ denote, respectively, the number of Sylow $p$-subgroups^{} and Sylow $q$-subgroups of $G$.

Sylow’s theorems tell us that ${n}_{q}=1+kq$ for some integer $k$ and ${n}_{q}$ divides $pq$. But $p$ and $q$ are prime and $$, so this implies that ${n}_{q}=1$. 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 $Q\cap P=\{1\}$ and $QP=G$, so $G$ is a semidirect product^{} of $Q$ and $P$. In particular, if there is only one Sylow $p$-subgroup, then $G$ is a direct product^{} of $Q$ and $P$, and is therefore cyclic.

Given $G=Q\u22caP$, it remains to determine the action of $P$ on $Q$ by conjugation^{}. There are two cases:

Case 1: If $p$ does not divide $q-1$, then since ${n}_{p}=1+mp$ cannot equal $q$ we must have ${n}_{p}=1$, and so $P$ is a normal subgroup^{} of $G$. This gives $G={C}_{p}\times {C}_{q}$ a direct product, which is isomorphic^{} to the cyclic group^{} ${C}_{pq}$.

Case 2: If $p$ divides $q-1$,
then $\mathrm{Aut}(Q)\cong {C}_{q-1}$ has a unique subgroup (http://planetmath.org/Subgroup) ${P}^{\prime}$ of order $p$,
where ${P}^{\prime}=\{x\mapsto {x}^{i}\mid i\in \mathbb{Z}/q\mathbb{Z},{i}^{p}=1\}$.
Let $a$ and $b$ be generators^{} for $P$ and $Q$ respectively,
and suppose the action of $a$ on $Q$ by conjugation is $x\mapsto {x}^{{i}_{0}}$,
where ${i}_{0}\ne 1$ in $\mathbb{Z}/q\mathbb{Z}$.
Then $G=\u27e8a,b\mid {a}^{p}={b}^{q}=1,ab{a}^{-1}={b}^{{i}_{0}}\u27e9$.
Choosing a different ${i}_{0}$
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 |