# subgroups with coprime orders

If the orders of two subgroups^{} of a group are coprime (http://planetmath.org/Coprime), the identity element^{} is the only common element of the subgroups.

*Proof.* Let $G$ and $H$ be such subgroups and $|G|$ and $|H|$ their orders. Then the intersection $G\cap H$ is a subgroup of both $G$ and $H$. By Lagrange’s theorem, $|G\cap H|$ divides both $|G|$ and $|H|$ and consequently it divides also $\mathrm{gcd}(|G|,|H|)$ which is 1. Therefore $|G\cap H|=1$, whence the intersection contains only the identity element.

Example. All subgroups

$$\{(1),(12)\},\{(1),(13)\},\{(1),(23)\}$$ |

of order 2 of the symmetric group^{} ${\U0001d516}_{3}$ have only the identity element $(1)$ common with the sole subgroup

$$\{(1),(123),(132)\}$$ |

of order 3.

Title | subgroups with coprime orders |
---|---|

Canonical name | SubgroupsWithCoprimeOrders |

Date of creation | 2013-03-22 18:55:58 |

Last modified on | 2013-03-22 18:55:58 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 20D99 |

Related topic | Gcd |

Related topic | CycleNotation |