# Difference between revisions of "Artinian group"

From Encyclopedia of Mathematics

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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]]]. | + | 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> | + | <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> | ||

## 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=11864

This article was adapted from an original article by V.P. Shunkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article