Order of an element of a group

The order of an element $x$ of a group $(G,\cdot)$ is the smallest positive integer $k$ such that $x^k=e$ where $e$ is the neutral element from $G$ . For instance, the order of $e$ is $1$ .

If this integer does not exist (in the case of infinite groups), we say that the order is infinite.

Practice this concept with exercises

Soit $(G,\cdot)$ un groupe fini d’élément neutre $e$ et $x\in G$ .
Montrer que l’application $n\in\Z\longmapsto x^n\in G$ n’est pas injective et en déduire qu’il existe un entier naturel $k$ tel que $x^k=e$ . Cela permet ainsi de définir l’ordre de $x$ .

Open exercise

Problems using this concept (1)

The group of units of ℤ/nℤ

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