Group of finite max-length

Definition
A group of finite max-length is a group satisfying the following equivalent conditions:


 * 1) The group is both a Noetherian group (i.e., it satisfies the ascending chain condition on all subgroups) and an Artinian group (i.e.,it satisfies the descending chain condition on all subgroups).
 * 2) Given any chain of subgroups of finite length, there is a chain of subgroups of finite length that refines it and that cannot be refined further.
 * 3) The max-length of the group is finite.

A group of finite max-length need not be finite, though counterexamples are rare. The best counterexamples are Tarski monsters, which have max-length two but are infinite.

Equivalence of definitions
The equivalence of definition relies on a Konig's lemma-style idea.