Difference between revisions of "Semi-simple group"
From Encyclopedia of Mathematics
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| − | A group whose [[ | + | 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> |
| − | + | <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> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , Hermann & 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> | |
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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
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