Order of an element

In additive notation the condition reads \(na=0\). In \((\mathbb{Z}_n,+)\) the element \(a\) has order \(n/\gcd(a,n)\); in \(S_n\), the order of a permutation is the least common multiple of the lengths of its disjoint cycles; in \((\mathbb{Z},+)\) every nonzero element has infinite order. The identity is the only element of order 1.

The key property is that \(a^k=e\) if and only if the order of \(a\) divides \(k\). The rest follows: \(|a^k|=|a|/\gcd(|a|,k)\), and \(a\) and \(a^{-1}\) have the same order. In a finite group every element has finite order, since the powers \(a,a^2,\dots\) cannot all be distinct.

A typical mistake is to believe that \(|ab|=|a|\,|b|\) or \(\operatorname{lcm}(|a|,|b|)\) always. This holds only if \(a\) and \(b\) commute and have coprime orders, in which case \(|ab|=|a|\,|b|\). In \(S_3\), two distinct transpositions of order 2 have a product of order 3.

The order of \(a\) equals the number of elements of the subgroup \(\{e,a,a^2,\dots\}\) it generates, an idea made formal by the generated subgroup and which, together with Lagrange, forces the order of an element to divide the order of the group.