= Order of a finite group
{title2=$|G|$}
= Group order
{synonym}
The <order of a finite group> is its <cardinality>, written $|G|$. The <identity element> is included: $|\mathbb Z/6\mathbb Z|=6$, and the <trivial group> has order one, rather than zero.
Keep this count distinct from the <order of a group element>. In the additive <finite cyclic group> $\mathbb Z/6\mathbb Z$, the element $1$ has order six, $2$ has order three, $3$ has order two, and $0$ has order one. Each element's order is the least positive number of repetitions needed to return to the <identity element> $0$. The <order of an element of a finite cyclic group> computes these periods directly.
The count also constrains <subgroups>: <Lagrange's theorem> gives $|G|=[G:H]|H|$ for a <subgroup> $H$ of a <finite group> $G$, where the <index of a subgroup> counts its <cosets>. For $H=\{0,2,4\}$ inside $\mathbb Z/6\mathbb Z$, the two <cosets> are $\{0,2,4\}$ and $\{1,3,5\}$, so $6=2\cdot3$. Divisibility is a necessary constraint; it does not imply that every divisor occurs as the <order of a group element>, as the <Klein four-group> shows.
Back to article page
