Let $G=D_{14}$ be the dihedral group of order 14 and let H be a subgroup of G. Suppose that $H\neq (e)$ and $H\neq G$. Prove that H is cyclic.
I'm confused on where to begin with this question, I understand that their exists some element that isn't the identity in the group and that the dihedral group is generated by 2 elements, one of order 7 and one of order 2. But how would I go about proving that H is cyclic?
Any hints/advice is much appreciated.
Best Answer
By Lagrange's theorem, any subgroup of $D_{14}$ must have order 1, 2, 7 or 14. Orders 1 and 14 are ruled out by the question, but the remaining possible orders are prime, so any subgroup of those orders must be cyclic, as required.