We say a group G is cyclic if there is an element g Element

We say a group G is cyclic if there is an element g Element G such that G = (g). If G = (g), we call g a generator of G. Show that every cyclic group is abelian. Let G be a finite group, and |G| = n. Show that G is cyclic if and only if G has an element of order n.

Solution

a)Let G be a cyclic group

And let g be the generaor of G.

Then any element of G should be some power of g.

Let g1,g2 belongs to G.

Then g1=g^(r) and g2=g^(s)

g1.g2=g^(r+s)=g^(s+r)=g^(s).g^(r)=g2.g1

Since g1,g2 are arbitrary,every elements of G satisfies this.

Hence G is abelian

b) If G is a cyclic group generated by g,then G=( g^(k) ) if and onlt if gcd(k,n)=1.thus G has an element of order n.

Conversly let G has an element g of order n,thrm g^(n)= identity.thus g^(n+1)=g itself..In similiar fashion we can express evry elements of G as some power of g.thus G is cuclic group generated by g.