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An abelian group is a group whose group operation is commutative: $gh=hg$ for every $g,h\in G$. Thus rearranging a finite sequence of operations preserves the result. Associativity already permits changing parentheses; commutativity additionally permits exchanging the elements.

The additive integers, the additive real numbers, and every cyclic group are abelian groups. Additive notation is often convenient: the identity element is written $0$, the inverse element of $a$ is $-a$, and $a+b=b+a$. It does not force an element to be an ordinary number: in $C_2\times C_2$, an element is a pair of binary coordinates, added componentwise modulo two.

Commutativity is useful because an integer linear combination of chosen group elements depends only on their coefficients, rather than on the order in which the operations were performed. The Fundamental theorem of finitely generated abelian groups turns this simplicity into a classification by cyclic groups. However, commutativity does not imply that a single generator of a group suffices: the Klein four-group is the smallest counterexample, as the cyclic group of order four and Klein four-group demonstrate.

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.