If G is a solvable group whose partially ordered set of subgroups satisfies both the ascending and descending chain conditions, then G is finite.

Question

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.

Answer 1

Establish the following facts:

• The result will follow if you can prove it for abelian groups.

• Since $G$ has ACC on subgroups, it follows that $G$ is finitely generated.

• Since $G$ has DCC on subgroups, it follows that $G$ is torsion.

• Torsion finitely generated abelian groups are finite.

For an example of a nonsolvable group that satisfies the condition but is not finite, look at Tarski monsters.