Order (group theory)

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Algebraic structure → Group theory Group theory
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In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that a^m = e, where e denotes the identity element of the group, and a^m denotes the product of m copies of a. If no such m exists, the order of a is infinite.

The order of a group G is denoted by ord(G) or |G|, and the order of an element a is denoted by ord(a) or |a|, instead of ${\displaystyle \operatorname {ord} (\langle a\rangle ),}$ where the brackets denote the generated group.

Lagrange's theorem states that for any subgroup H of a finite group G, the order of the subgroup divides the order of the group; that is, |H| is a divisor of |G|. In particular, the order |a| of any element is a divisor of |G|.