View source for Artinian group

''group with the minimum condition for subgroups, group with the descending chain condition''

A group in which any decreasing chain of distinct subgroups terminates after a finite number. Artinian groups are periodic, and the question of their structure hinges on Schmidt's problem on infinite groups with finite proper subgroups [[#References|[3]]] and the minimality problem: Is an Artinian group a finite extension of an Abelian group? Both these problems have been solved for locally solvable groups [[#References|[1]]] and locally finite groups [[#References|[3]]], [[#References|[4]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Chernikhov,   "Infinite locally solvable groups"  ''Mat. Sb.'' , '''7 (49)''' :  1  (1940)  pp. 35–64  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Chernikhov,   "The finiteness condition in general group theory"  ''Uspekhi Mat. Nauk'' , '''14''' :  5  (1959)  pp. 45–96  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Kargapolov,   "On a problem of O.Yu. Schmidt"  ''Sibirsk. Mat. Zh.'' , '''4''' :  1  (1963)  pp. 232–235  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.P. Shunkov,   "On the minimality property for locally finite groups"  ''Algebra and Logic'' , '''9''' :  2  (1970)  pp. 137–151  ''Algebra i Logika'' , '''9''' :  2  (1970)  pp. 220–248</TD></TR></table>



====Comments====
Schmidt's problem actually states: Under what conditions does an infinite group have proper infinite subgroups?

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O.H. Kegel,   B.V. Wehrfritz,   "Locally finite groups" , North-Holland  (1973)</TD></TR></table>