= Finite group
{wiki}
A <finite group> is a <group> whose underlying <set> has finitely many elements. Its <order of a finite group> is the number of elements, counting the <identity element>. The <group operation> still has to satisfy all the <group axioms>: merely having a finite list of states does not make a <group>.
For example, the six <congruence classes> modulo six form a <finite cyclic group> under <addition>, whereas the <permutations> of three objects form the <symmetric group> $S_3$, also with six elements. The first is an <abelian group>; the second is a <non-abelian group>. Thus the <order of a finite group> is a useful first measurement, but the composition rule contains information that the count alone loses.
Finiteness makes <group> structure accessible to exact calculation. A <Cayley table> for a <finite group> of order $n$ has $n^2$ entries, and every <group element> eventually returns to the <identity element> when repeated. <Lagrange's theorem> strengthens this observation: the <order of a group element>, being the size of its <cyclic subgroup>, divides $n$. The <cyclic group of order four and Klein four-group> give a small example of how that constraint can distinguish different composition rules.
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