In group theory the use of the phrase semi-simple group is used sparingly. Standard texts on group theory including [1, 2] avoid the term altogether. Other texts provide precise definitions which are nevertheless not equivalent [3, 4]. In general it is preferable to use other terms to describe the class of groups being considered as there is no uniform convention. However, below is a list of possible uses of for the phrase semi-simple group.
A product of simple groups may be called semi-simple. Depending on application, the simple groups may be further restricted to finite simple groups and may also exclude the abelian simple groups.
A Lie group whose associated Lie algebra is a semi-simple Lie algebra may be called a semi-simple group and more specifically, a semi-simple Lie group.
The use of semi-simple in the study of algebras, representation theory, and modules is far more precise owing to the fact that the various possible definitions are generally equivalent.
For example. In a finite dimensional associative algebra , if it is a product of simple algebras then the Jacobson radical is trivial. In contrast, if has trivial Jacobson radical then it is a direct product of simple algebras. Thus may be called semi-simple if either: is a direct product of simple algebras or has trivial Jacobson radical.
The analogue fails for groups. For instance. If a group is defined as semi-simple by virtue of having no non-trivial normal abelian subgroups then is semi-simple for all . However, is not a product of simple groups.
- 1 Aschbacher, M. Finite group theory Cambridge studies in advanced mathematics 10, Cambridge University Press, Cambridge, (1986).
- 2 Gorenstein, D. Finite groups Chelsea Publishing Company, New York, (1980).
- 3 Robinson, D. J.S. A course in the theory of groups Ed. 2, GTM 80, Springer, New York, (1996).
- 4 Suzuki, M. Group Theory I,II, (English) Springer-verlag, Berlin (1982, 1986).
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