A group $G$ is said cocyclic if has a non-trivial minimum subgroup M or,equivantly, if the intersection M of all non-trivial subgroups of G is non-trivial. For exemple the quaternion group $Q_8$ is cocyclic.
I have found this caraterizaction:
Let $G$ a abelian group, then are equivalent:
$G$ is cocyclic.
the lattice of subgroup of $G$ is a chain.
$G$ is isomorphic to $\mathbb{Z}(p^k)$ with $k\in\{1,2,...\}\cup\{\infty\}$ and $p$ prime.
The proof is based on the fact 'A periodic abelian indecomposable group is isomorph to to $\mathbb{Z}(p^k)$ with $k\in\{1,2,...\}\cup\{\infty\}$ and $p$ prime'.
Now i ask if the hypotesis '$G$ is abelian' is necessary in $2)\Longrightarrow 3)$. In particular i ask if a group with lattice totally ordered is abelian or exists a non abelian counterexample.
Thanks in advance.
If $G$ is nonabelian then $gh\neq hg$ for some $g,h\in G$. Then clearly neither $\langle g\rangle\subseteq \langle h\rangle$ nor $\langle h\rangle\subseteq \langle g\rangle$ since cyclic groups are abelian. And so the lattice of subgroups is not totally ordered.