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Difference between revisions of "Artinian group"

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''group with the minimum condition for subgroups, group with the descending chain condition''
 
''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]]].
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A group in which any decreasing chain of distinct subgroups terminates after a finite number. Artinian groups are [[Periodic group|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====
 
====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>
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<table>
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<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>
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<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>
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<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>
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<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>
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</table>
  
  

Latest revision as of 20:29, 17 October 2014

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 [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 [1] and locally finite groups [3], [4].

References

[1] S.N. Chernikhov, "Infinite locally solvable groups" Mat. Sb. , 7 (49) : 1 (1940) pp. 35–64 (In Russian)
[2] S.N. Chernikhov, "The finiteness condition in general group theory" Uspekhi Mat. Nauk , 14 : 5 (1959) pp. 45–96 (In Russian)
[3] M.I. Kargapolov, "On a problem of O.Yu. Schmidt" Sibirsk. Mat. Zh. , 4 : 1 (1963) pp. 232–235 (In Russian)
[4] 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


Comments

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

References

[a1] O.H. Kegel, B.V. Wehrfritz, "Locally finite groups" , North-Holland (1973)
How to Cite This Entry:
Artinian group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Artinian_group&oldid=33742
This article was adapted from an original article by V.P. Shunkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article