Order of a group

The order \(|G|\) of a group \(G\) is the size of its underlying set. For example, if \(G=(X,\bullet)\) and \(X\) has nine elements, we say that \(G\) has order \(9\). If \(X\) is infinite, we say \(G\) is infinite; if \(X\) is finite, we say \(G\) is finite.

The order of an element \(g \in G\) of a group is the smallest nonnegative integer \(n\) such that \(g^n = e\), or \(\infty\) if there is no such integer. The relationship between this usage of order and the above usage of order is that the order of \(g \in G\) in this sense is the order of the subgroup \(\langle g \rangle = \{ 1, g, g^2, \dots \}\) of \(G\) generated by \(g\) in the above sense.


  • Group

    The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.