= Abelian group
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= Abelian
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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.
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