Noetherian group

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group with the maximum condition for subgroups

A group in which every strictly ascending chain of subgroups is finite. This class is named after E. Noether, who investigated rings with the maximum condition for ideals — Noetherian rings (cf. Noetherian ring). Subgroups and quotient groups of a Noetherian group are Noetherian. Examples have been constructed of Noetherian groups that are not finite extensions of polycyclic groups (cf. Polycyclic group) [1].


[1] A.Yu. Ol'shanskii, "Infinite groups with cyclic subgroups" Soviet Math. Dokl. , 20 : 2 (1979) pp. 343–346 Dokl. Akad. Nauk SSSR , 245 : 4 (1979) pp. 785–787
How to Cite This Entry:
Noetherian group. V.N. Remeslennikov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098