Prove that every cyclic group is abelian.

Prove that every cyclic group is abelian.

Ans.

    Let, G be a cyclic group generated by a.

    Let, x, y∈G be two arbitrary elements.

∴ ∃ some m∈z s.t. x=am

& ∃ some m∈z s.t. y =an

Now, x.y = am.an = an+m   [ ∵ m,n∈z = m+n = n+m ]

                              = an.am

                              = y.x

    ∴  [  x.y = y.x  ]

Since, x, y were arbitrary

        So, it holds for all x,y∈G

∴ G is an abelian group.