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Concept
Order (group theory)
Summary
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 am = e, where e denotes the identity element of the group, and am 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 , and the order of an element a is denoted by ord(a) or , instead of \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, is a divisor of . In particular, the order of any elem
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