The problem statement, all variables and given/known data If H ≤ G is cyclic and normal in G, prove that every subgroup of H is also normal in G. The attempt at a solution Let H = <h>. We know that for g in G, hi = ghjg-1 by the normality of H. A simple induction shows that hin = ghjng-1, so that <hi> = g<hj>g-1. Now all I need to show is that hj belongs to <hi>. I'm having trouble with this. Any tips?