Editing Subgroup series (section)

===Noetherian groups, Artinian groups===
A group that satisfies the [[ascending chain condition]] (ACC) on subgroups is called a '''Noetherian group''', and a group that satisfies the [[descending chain condition]] (DCC) is called an '''Artinian group''' (not to be confused with [[Artin group]]s), by analogy with [[Noetherian ring]]s and [[Artinian ring]]s. The ACC is equivalent to the '''maximal condition''': every [[Empty set|non-empty]] collection of subgroups has a maximal member, and the DCC is equivalent to the analogous '''minimal condition'''.

A group can be Noetherian but not Artinian, such as the [[infinite cyclic group]], and unlike for [[Ring (mathematics)|rings]], a group can be Artinian but not Noetherian, such as the [[Prüfer group]]. Every finite group is clearly Noetherian and Artinian.

[[Group homomorphism|Homomorphic]] [[Image (mathematics)|images]] and subgroups of Noetherian groups are Noetherian, and an [[group extension|extension]] of a Noetherian group by a Noetherian group is Noetherian. Analogous results hold for Artinian groups.

Noetherian groups are equivalently those such that every subgroup is [[finitely generated group|finitely generated]], which is stronger than the group itself being finitely generated: the [[free group]] on 2 or finitely more generators is finitely generated, but contains free groups of infinite rank.

Noetherian groups need not be finite extensions of [[polycyclic group]]s.<ref>{{cite journal | author = Ol'shanskii, A. Yu. | year = 1979 | title = Infinite Groups with Cyclic Subgroups | journal = Soviet Math. Dokl. | volume = 20 | pages = 343&ndash;346}} (English translation of ''Dokl. Akad. Nauk SSSR'', '''245''', 785&ndash;787)</ref>