Order of a group

It is written \(|G|\). In \((\mathbb{Z}_n,+)\) it equals \(n\); in the symmetric group \(S_n\) of permutations of \(n\) objects it equals \(n!\); and in the group of units mod \(n\), made of the residues coprime to \(n\), it equals \(\varphi(n)\), Euler's function. The trivial group \(\{e\}\) has order 1, the minimum possible, because every group contains its identity. Groups like \((\mathbb{Z},+)\) or \((\mathbb{R}^*,\cdot)\) have infinite order.

The order is the first invariant: if two groups have different orders, they cannot have the same structure. The converse is false, since there are different groups of the same order: \(\mathbb{Z}_4\) and the group of units mod 12 both have 4 elements, but in the first one element generates everything and in the second none does.

The order constrains the subgroups. Lagrange's theorem, which we will see later, states that the order of any subgroup of a finite group divides the order of the group. For example, \(S_3\), of order 6, has no subgroups of order 4 or 5. This is why groups of prime order are especially simple.

Do not confuse the order of the group, \(|G|\), with the order of an element, which is the least \(n>0\) with \(a^n=e\) and is introduced next. They are related: the order of an element equals the order of the subgroup it generates.