Definition of Cyclic subgroup 


The above is a theorem from my book. What I don't understand is the second sentence when it says $b$ generates $H$ with $n/d$ elements. I thought that since $b = a^s$ generates $H$, it would have $s$ elements, meaning $H = \{e, a, a^2, \dots, a^s \}$? I've found some counterexamples to convince myself I am wrong, but I don't understand why my original idea is wrong either. 

Another proof
Let $k=o\langle b\rangle=o\langle a^s\rangle$ be the order of H
and according to the definition,
$$a^n=e$$
and
$$b^k=(a^s)^k=a^{ks}=e$$
so that
$$n\mid ks$$
where k is the minimum possible value
$$ks\equiv 0 \pmod n$$
and finally
$$k=\frac{n}{\text{gcd}(n,s)}=n/d$$
