Let G be a finite group and let H be a normal subgroup of G

Let G be a finite group and let H be a normal subgroup of G. Prove that the order of the element g H in G/H must divide the order of g in G.

Solution

Let order of g in G be n.

Also the identity of G/H is H itself.

Now consider (gH)ⁿ,

Thus,

(gH)ⁿ = (gⁿ)H = eH = H

Therefore, the order of gH divides order of g in G.