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On Primitive Solvable Linear Groups

Published online by Cambridge University Press:  20 November 2018

C. R. B. Wright
C. R. B. Wright*
Affiliation:
University of Oregon, Eugene, Oregon
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Let V be a vector space over the field K. A group G of K-linear transformations of V onto itself is primitive in case no proper nontrivial subspace of V is G-invariant and V cannot be written as a direct sum of proper subspaces permuted among themselves by G. Equivalently, G is primitive on V in case G is irreducible and is not induced from a proper subgroup.

Suprunenko showed [3, Theorem 12, p. 28] that the n-dimensional general linear group GL(n, K) has a solvable primitive subgroup only if

(1) there is a divisor, m, of n such that K has an extension field of degree m containing a primitive p-th root of 1 for each prime p dividing n/m.

The main result of this note is the converse fact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 679 - 685
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Artin, E., Nesbitt, C. J. and Thrall, R. M., Rings with minimum condition (University of Michigan Press, Ann Arbor, 1946).Google Scholar
2. Huppert, B., Endliche Gruppen I (Springer Verlag, Berlin-Heidelberg-New York, 1967).Google Scholar
3. Suprunenko, D., Soluble and nilpotent linear groups, Translations of Mathematical Monographs No. 9 (Amer. Math. Soc, Providence, 1963).Google Scholar