Finite Group whose Subsets form Nest is Cyclic P-Group

Theorem
Let $G$ be a group.

Let $G$ be such that its subgroups form a nest.

Then $G$ is a cyclic $p$-group.

Proof
Suppose $G$ is not a $p$-group.

Then there exist two distinct primes $p_1, p_2$.

By Cauchy's Group Theorem, there exist subgroups $H, K$ such that:
 * $\order H = p_1$
 * $\order K = p_2$

That is:
 * $H = \gen a$
 * $k = \gen b$

for some $a, b \in G: a \ne b$, where:
 * $\order a = p_1$
 * $\order b = p_2$

and so both $H \nsubseteq K$ and $K \nsubseteq H$

Thus, by definition, the subgroups of $G$ do not form a nest.

It follows by the Rule of Transposition that $G$ is a $p$-group.

It remains to be shown that $G$ is cyclic.

Suppose to the contrary that $G$ is not cyclic.

Then it is not the case that $G$ is generated by a single element of $G$.

Thus there exist $a, b \in G, a \ne b$ such that $H = \gen a$ and $K = \gen b$ are subgroups of $G$.

But as $a \in H, b \in K, a \notin K, b \notin H$ it follows that $H \nsubseteq K$ and $K \nsubseteq H$.

It follows by the Rule of Transposition that $G$ is cyclic.