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text: '[summary: An abelian group is a [3gd group] where the operation is [3jb commutative]. That is, an abelian group $G$ is a pair $(X, \\bullet)$ where $X$ is a [3jz set] and $\\bullet$ is a binary [3h7 operation] obeying the four group axioms plus an axiom of commutativity:\n\n1. [3gy Closure]: For all $x, y$ in $X$, $x \\bullet y$ is defined and in $X$. We abbreviate $x \\bullet y$ as $xy$.\n2. [3h4 Associativity]: $x(yz) = (xy)z$ for all $x, y, z$ in $X$.\n3. Identity: There is an element $e$ such that for all $x$ in $X$, $xe=ex=x$.\n4. Inverses: For each $x$ in $X$ is an element $x^{-1}$ in $X$ such that $xx^{-1}=x^{-1}x=e$.\n5. [3jb Commutativity]: For all $x, y$ in $X$, $xy=yx$.\n \nAbelian groups are very "well-behaved" groups that are often easier to deal with than their non-commuting counterparts.]\n\nAn abelian group is a [3gd group] $G=(X, \\bullet)$ where $\\bullet$ is [3jb commutative]. In other words, the group operation satisfies the five axioms:\n\n1. [3gy Closure]: For all $x, y$ in $X$, $x \\bullet y$ is defined and in $X$. We abbreviate $x \\bullet y$ as $xy$.\n2. [3h4 Associativity]: $x(yz) = (xy)z$ for all $x, y, z$ in $X$.\n3. Identity: There is an element $e$ such that for all $x$ in $X$, $xe=ex=x$.\n4. Inverses: For each $x$ in $X$ is an element $x^{-1}$ in $X$ such that $xx^{-1}=x^{-1}x=e$.\n5. [3jb Commutativity]: For all $x, y$ in $X$, $xy=yx$.\n\nThe first four are the standard [3gd group axioms]; the fifth is what distinguishes abelian groups from groups. \n\nCommutativity gives us license to re-arrange chains of elements in formulas about commutative groups. For example, if in a commutative group with elements $\\{1, a, a^{-1}, b, b^{-1}, c, c^{-1}, d\\}$, we have the claim $aba^{-1}db^{-1}=d^{-1}$, we can shuffle the elements to get $aa^{-1}bb^{-1}d=d^{-1}$ and reduce this to the claim $d=d^{-1}$. This would be invalid for a nonabelian group, because $aba^{-1}$ doesn't necessarily equal $aa^{-1}b$ in general.\n\nAbelian groups are very well-behaved groups, and they are often much easier to deal with than their non-commutative counterparts. For example, every [576] of an abelian group is [4h6 normal], and all finitely generated abelian groups are a [group_theory_direct_product direct product] of [47y cyclic groups] (the [structure_theorem_for_finitely_generated_abelian_groups structure theorem for finitely generated abelian groups]). ',
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