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Homesubgroups with coprime orders

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# subgroups with coprime orders

If the orders of two subgroups of a group are 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 $\gcd(|G|,\,|H|)$ which is 1. Therefore $|G\!\cap\!H|=1$, whence the intersection contains only the identity element.

Example. All subgroups

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

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

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

of order 3.

Related:

Gcd, CycleNotation

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

20D99*no label found*

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