# Abelian group

An abelian group is a group $$G=(X, \bullet)$$ where $$\bullet$$ is com­mu­ta­tive. In other words, the group op­er­a­tion satis­fies the five ax­ioms:

1. Clo­sure: For all $$x, y$$ in $$X$$, $$x \bullet y$$ is defined and in $$X$$. We ab­bre­vi­ate $$x \bullet y$$ as $$xy$$.

2. As­so­ci­a­tivity: $$x(yz) = (xy)z$$ for all $$x, y, z$$ in $$X$$.

3. Iden­tity: There is an el­e­ment $$e$$ such that for all $$x$$ in $$X$$, $$xe=ex=x$$.

4. In­verses: For each $$x$$ in $$X$$ is an el­e­ment $$x^{-1}$$ in $$X$$ such that $$xx^{-1}=x^{-1}x=e$$.

5. Com­mu­ta­tivity: For all $$x, y$$ in $$X$$, $$xy=yx$$.

The first four are the stan­dard group ax­ioms; the fifth is what dis­t­in­guishes abelian groups from groups.

Com­mu­ta­tivity 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.

Abe­lian groups are very well-be­haved groups, and they are of­ten much eas­ier to deal with than their non-com­mu­ta­tive coun­ter­parts. For ex­am­ple, ev­ery Sub­group of an abelian group is nor­mal, and all finitely gen­er­ated abelian groups are a di­rect product of cyclic groups (the struc­ture the­o­rem for finitely gen­er­ated abelian groups).

Parents:

• Group

The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.

• Algebraic structure
• I strongly recom­mend keep­ing to the stan­dard term “abelian group,” even though “com­mu­ta­tive group” would be more sys­tem­atic and sen­si­ble. The term “abelian group” is uni­ver­sal—I don’t know a sin­gle math­e­mat­i­cian, book, or pa­per that uses the term “com­mu­ta­tive group”—and peo­ple com­par­ing what they read here to what they read any­where else are just go­ing to be con­fused, and/​or are go­ing to con­fuse third par­ties when they ask ques­tions.