Geodesic
Solvable and Nilpotent Subgroups of GL(n,q m )
Canadian journal of mathematics, Tome 34 (1982) no. 5, pp. 1097-1111

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

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| ≦ q 3nm = |V|3. In fact, we show that where 
Wolf, Thomas R. Solvable and Nilpotent Subgroups of GL(n,q m ). Canadian journal of mathematics, Tome 34 (1982) no. 5, pp. 1097-1111. doi: 10.4153/CJM-1982-079-5
@article{10_4153_CJM_1982_079_5,
     author = {Wolf, Thomas R.},
     title = {Solvable and {Nilpotent} {Subgroups} of {GL(n,q} m )},
     journal = {Canadian journal of mathematics},
     pages = {1097--1111},
     year = {1982},
     volume = {34},
     number = {5},
     doi = {10.4153/CJM-1982-079-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-079-5/}
}
TY  - JOUR
AU  - Wolf, Thomas R.
TI  - Solvable and Nilpotent Subgroups of GL(n,q m )
JO  - Canadian journal of mathematics
PY  - 1982
SP  - 1097
EP  - 1111
VL  - 34
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-079-5/
DO  - 10.4153/CJM-1982-079-5
ID  - 10_4153_CJM_1982_079_5
ER  - 
%0 Journal Article
%A Wolf, Thomas R.
%T Solvable and Nilpotent Subgroups of GL(n,q m )
%J Canadian journal of mathematics
%D 1982
%P 1097-1111
%V 34
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-079-5/
%R 10.4153/CJM-1982-079-5
%F 10_4153_CJM_1982_079_5

[1] 1. Burnside, W., On groups of order paqb, Proc. London Math. Soc. 2 (1904), 388–392./ Google Scholar

[2] 2. Burnside, W., On groups of order paqh II, Proc. London Math. Soc. 2 (1904), 432–437./ Google Scholar

[3] 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] 4. Dixon, J., The Fitting subgroup of a linear solvable group, J. Australian Math. Soc. 7 (1967), 417–424./ Google Scholar

[5] 5. Dixon, J., Normal p-subgroups of solvable linear groups, J. Australian Math. Soc. 7 (1967), 545–551./ Google Scholar

[6] 6. Dixon, J., The structure of linear groups (Van Nostrand Reinhold, London, 1971./ Google Scholar

[7] 7. Glauberman, G., On Burnside's other paqb theorem, Pac. J. Math. 56 (1975), 469–475./ Google Scholar

[8] 8. Gorenstein, D., Finite groups (Harper and Row, New York, 1968./ Google Scholar

[9] 9. Huppert, B., Lineare aufslôbare gruppen, Math. Z. 67 (1957), 479–518./ Google Scholar

[10] 10. Huppert, B., Endliche gruppen I (Springer-Verlag, Berlin, 1967./ Google Scholar

[11] 11. Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976./ Google Scholar

[12] 12. Isaacs, I. M., Character correspondences in solvable groups, Advances in Math. 43 (1982), 284–306./ Google Scholar

[13] 13. Passman, D., Permutation groups (Benjamin, New York, 1968./ Google Scholar

[14] 14. Suprunenko, D., Soluble and nilpotent linear groups (A.M.S., Providence, 1936./ Google Scholar

Cité par Sources :