Suppose $G$ is a solvable group whose partially ordered set of subgroups satisfies both the ascending and descending chain conditions.
GOAL: Show that $G$ is finite.
- Since $G$ is solvable, there exists a finite collection of normal subgroups in $G$ such that $1 = N_0 \le N_1 \le \ldots \le N_n=G$ and the factor groups are all abelian.
- Let $P$ be the poset of subgroups of $G$. Since $P$ satisfies both the ACC and DCC, every ascending chain and descending chain in $P$ is 'eventually constant'.
- We know $P$ satisfies the ACC $\iff$ every nonempty subset of $P$ has a maximal element. Likewise, $P$ satisfies the DCC $\iff$ every nonempty subset of $P$ has a minimal element.
Thank you for your help.