Geodesic
On Primitive Solvable Linear Groups
Canadian journal of mathematics, Tome 23 (1971) no. 4, pp. 679-685

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Wright, C. R. B. On Primitive Solvable Linear Groups. Canadian journal of mathematics, Tome 23 (1971) no. 4, pp. 679-685. doi: 10.4153/CJM-1971-075-6
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[1] 1. Artin, E., Nesbitt, C. J. and Thrall, R. M., Rings with minimum condition (University of Michigan Press, Ann Arbor, 1946). Google Scholar

[2] 2. Huppert, B., Endliche Gruppen I (Springer Verlag, Berlin-Heidelberg-New York, 1967). Google Scholar

[3] 3. Suprunenko, D., Soluble and nilpotent linear groups, Translations of Mathematical Monographs No. 9 (Amer. Math. Soc, Providence, 1963). Google Scholar

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