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Complete Reductibility of Infinite Groups

Published online by Cambridge University Press:  20 November 2018

John D. Dixon
John D. Dixon*
Affiliation:
California Institute of Technology
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The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.

It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:

If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 267 - 274
Copyright
Copyright © Canadian Mathematical Society 1964

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