Order (group theory)

In group theory, order can mean the cardinality of a group or the least positive

n

for which a group element

g

satisfies

g^{n} = e

. If no such

n

exists, the element has infinite order. For a finite group, Lagrange's theorem implies that each element order divides the group order.

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Order of an element of a finite group

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In the context of group theory, the term "order" can refer to two related but distinct concepts: 1. **Order of a Group**: The order of a group \( G \), denoted as \( |G| \), is defined as the number of elements in the group. For finite groups, this is simply a count of all the elements.

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