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