Abelian Group

An abelian group is a Group in which the multiplication operation is Commutative.

Properties

• Abelian groups are equivalent to Modules over the Ring of Integers.

  To start, note that modules are abelian groups by definition, so it suffices to show that every abelian group $G,+,0$ is a $\mathbb{Z}$-module.

  Consider the operation $(x,n)\mapsto nx$ for $x:G,n:\mathbb{Z}$. We need to show:

  • $(m+n)x=mx+nx$
  • $(mn)x=m(nx)$
  • $1x=x$
  • $m(x+y)=mx+my$

  These all follow from some easy case splits and induction.