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Difference between revisions of "Semi-simple group"

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''(in the sense of some radical)''
 
''(in the sense of some radical)''
  
A group whose [[Radical|radical]] is the identity subgroup. Thus, the concept of a semi-simple group is entirely defined by the choice of a radical class of groups. In the theory of finite groups and Lie groups, by a radical one usually understands a maximal (connected) solvable normal subgroup. In these cases, the description of semi-simple groups is essentially reduced to the description of simple groups.
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A group whose [[radical]] is the identity subgroup. Thus, the concept of a semi-simple group is entirely defined by the choice of a radical class of groups. In the theory of finite groups and Lie groups, by a radical one usually understands a maximal (connected) solvable normal subgroup. In these cases, the description of semi-simple groups is essentially reduced to the description of simple groups.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann &amp; Masson  (1960–1982)  pp. Chapts. I-IX</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Hochschild,  "The structure of Lie groups" , Holden-Day  (1965)</TD></TR>
 
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</table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann &amp; Masson  (1960–1982)  pp. Chapts. I-IX</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Hochschild,  "The structure of Lie groups" , Holden-Day  (1965)</TD></TR></table>
 

Latest revision as of 14:09, 15 April 2023

(in the sense of some radical)

A group whose radical is the identity subgroup. Thus, the concept of a semi-simple group is entirely defined by the choice of a radical class of groups. In the theory of finite groups and Lie groups, by a radical one usually understands a maximal (connected) solvable normal subgroup. In these cases, the description of semi-simple groups is essentially reduced to the description of simple groups.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[a1] N. Bourbaki, "Groupes et algèbres de Lie" , Hermann & Masson (1960–1982) pp. Chapts. I-IX
[a2] G. Hochschild, "The structure of Lie groups" , Holden-Day (1965)
How to Cite This Entry:
Semi-simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_group&oldid=53797
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article