be a group. [tex]a,b /in G[\math]

Order of

Order of

Prove there exists an element in with order = LCM . Let this LCM be

My approach:

Consider

Let order of be

Obviously (as )

Only two cases are possible

Case 1:AND- we are done here

Case 2: doesn't divideANDdoesn't divide

In this case we have = where and

Stuck here for quite some time.

Is my approach correct? If yes, hint on way fwd please?