Let G be a group with n elements a b elementof G and the set

Let G be a group with n elements, a, b elementof G and the set S = {c elementof G/ac = cb). Let k be the number of elements of S. Prove that either k = 0, or k divides n.

Solution

Here we have given that ac=cb

or ac-cb =0

c(a-b)=0

So either c= 0 or a-b=0 i.e. a=b

Now if S have total k elements, so by c=0, it is cleared that total elements k of S be also 0 or k=0.

Now again as a,b,c are the elements of G, so clearly

pa+qb+rc = n or pa+qa + rk = n

or rk=n-a(p-q)

or k = (n-a(p-q))/r (where p,q and r be any scalar)

that means k divides n.

Thus either k=0 or k divides n.