Let G be a cyclic p-group with subgroups H and K. Prove that either H is contained in K or K is contained in H.

Question

I am working on this question for Abstract Algebra (undergraduate level) and I think I may have to use Lagrange's theorem? Not sure if I'm even on the right track, and if I am I'm not sure where to go from there.

Answer 1

Write $G=\langle x \rangle$ where $|x|=p^n$.
Suppose that $H\neq K.$
Then $H=\langle x^{p^\alpha}\rangle$ and $K=\langle x^{p^\beta}\rangle$ where $0\leq\alpha<\beta\leq n$.

Since $p^\alpha\mid p^\beta$, write $p^\beta=cp^\alpha$ for some intger $c$.
Then $x^{p^\beta}=(x^{p^\alpha})^c\in H$
So $H\leq K$.

Important Fact
If $G$ is a cyclic group of order $n$ which is generated by $x$, then for every divisor $d$ of $n$, there is exactly one subgroup $H$ of order $d$, which can be generated by $x^{n/d}$.