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Solvable and Nilpotent Subgroups of GL(n,qm)

Published online by Cambridge University Press:  20 November 2018

Thomas R. Wolf
Thomas R. Wolf*
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin
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Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.

Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show that

where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 5 , 01 October 1982 , pp. 1097 - 1111
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Burnside, W., On groups of order paqb, Proc. London Math. Soc. 2 (1904), 388–392./Google Scholar
2. Burnside, W., On groups of order paqh II, Proc. London Math. Soc. 2 (1904), 432–437./Google Scholar
3. Coates, M., Dwan, M. and Rose, J. S., A note on Burnside's otherp aqP theorem, J. London Math. Soc. (2) (1976), 160–166./Google Scholar
4. Dixon, J., The Fitting subgroup of a linear solvable group, J. Australian Math. Soc. 7 (1967), 417–424./Google Scholar
5. Dixon, J., Normal p-subgroups of solvable linear groups, J. Australian Math. Soc. 7 (1967), 545–551./Google Scholar
6. Dixon, J., The structure of linear groups (Van Nostrand Reinhold, London, 1971./Google Scholar
7. Glauberman, G., On Burnside's other paqb theorem, Pac. J. Math. 56 (1975), 469–475./Google Scholar
8. Gorenstein, D., Finite groups (Harper and Row, New York, 1968./Google Scholar
9. Huppert, B., Lineare aufslôbare gruppen, Math. Z. 67 (1957), 479–518./Google Scholar
10. Huppert, B., Endliche gruppen I (Springer-Verlag, Berlin, 1967./Google Scholar
11. Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976./Google Scholar
12. Isaacs, I. M., Character correspondences in solvable groups, Advances in Math. 43 (1982), 284–306./Google Scholar
13. Passman, D., Permutation groups (Benjamin, New York, 1968./Google Scholar
14. Suprunenko, D., Soluble and nilpotent linear groups (A.M.S., Providence, 1936./Google Scholar