The Complete Reducibility of Locally Completely Reducible Finitary Linear Groups

1997 ◽  
Vol 29 (2) ◽  
pp. 173-176 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Groups ◽  
Complete Reducibility ◽  
Finitary Linear ◽  
Completely Reducible

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

scholarly journals COMPLETELY REDUCIBLE SETS

2013 ◽  
Vol 23 (04) ◽  
pp. 915-941 ◽  
Author(s):  
DOMINIQUE PERRIN
Keyword(s):  
Closure Properties ◽  
Linear Representations ◽  
Complete Reducibility ◽  
The Family ◽  
Syntactic Representation ◽  
Completely Reducible ◽  
Rational Sets ◽  
Cyclic Sets

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

Locally Solvable Finitary Linear Groups

1993 ◽  
Vol s2-47 (1) ◽  
pp. 31-40 ◽  
Author(s):  
Ulrich Meierfrankenfeld ◽  
Richard E. Phillips ◽  
Orazio Puglisi
Keyword(s):  
Linear Groups ◽  
Finitary Linear

Unipotent Finitary Linear Groups

1993 ◽  
Vol s2-48 (1) ◽  
pp. 59-76 ◽  
Author(s):  
Felix Leinen ◽  
Orazio Puglisi
Keyword(s):  
Linear Groups ◽  
Finitary Linear

scholarly journals Finitarily linear wreath products

2000 ◽  
Vol 43 (1) ◽  
pp. 27-41
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Finite Dimension ◽  
Wreath Products ◽  
Vector Spaces ◽  
Normal Subgroups ◽  
Infinite Dimensional ◽  
Linear Representations ◽  
Finitary Linear ◽  
Completely Reducible ◽  
Necessary And Sufficient ◽  
Standard Wreath Product

AbstractWe consider faithful finitary linear representations of (generalized) wreath products A wrΩH of groups A by H over (potentially) infinite-dimensional vector spaces, having previously considered completely reducible such representations in an earlier paper. The simpler the structure of A the more complex, it seems, these representations can become. If A has no non-trivial abelian normal subgroups, the conditions we present are both necessary and sufficient. They imply, for example, that for such an A, if there exists such a representation of the standard wreath product A wr H of infinite dimension, then there already exists one of finite dimension.

scholarly journals Coprime subdegrees for primitive permutation groups and completely reducible linear groups

2012 ◽  
Vol 195 (2) ◽  
pp. 745-772 ◽  
Author(s):  
Silvio Dolfi ◽  
Robert Guralnick ◽  
Cheryl E. Praeger ◽  
Pablo Spiga
Keyword(s):  
Permutation Groups ◽  
Linear Groups ◽  
Completely Reducible
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