Order (Group Theory)

In the context of group theory, the notion of an order means a slightly different thing when talking about a group when compared to an element of a group.

Order (Element)

Given an element \(g\) in a group \(G\), the order of \(g\), denoted by \(\mathrm{ord}(g)\), is the smallest positive integer \(n\) such that

\[ g^{n} = \mathrm{id}.\]

Such an element may not exist, in which case we say the order is infinite.

Order (Group)

The order of the group \(G\) is the number of elements in \(G\) if \(G\) is finite.