Finite Linear Groups of Degree Six

II, J. H. Lindsey

Canadian journal of mathematics, Tome 23 (1971) no. 5, pp. 771-790

Résumé

In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).

Détail

DOI : 10.4153/CJM-1971-086-x

Bibliographie

[1] 1. Blichfeldt, H. F., Finite collineation groups (University of Chicago Press, Chicago, 1917). Google Scholar

[2] 2. Brauer, R., On the connection between the ordinary and modular characters of groups of finite order, Ann. of Math. 42 (1941), 926–935. Google Scholar

[3] 3. Brauer, R., Investigations on group characters, Ann. of Math. 42 (1941), 936–958. Google Scholar

[4] 4. Brauer, R., On groups whose order contains a prime number to the first power. I, II, Amer. J. Math. 64 (1942), 401-420, 421–440. Google Scholar

[5] 5. Brauer, R., Uber endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 73–96. Google Scholar

[6] 6. Brauer, R., On simple groups of order 5-3a * 2b (Dept. Math, mimeographed notes, Harvard University, 1967). Google Scholar

[7] 7. Brauer, R., On a theorem of Burnside, Illinois J. Math. II (1967), 349–352. Google Scholar

[8] 8. Brauer, R. and Leonard, H. S., Jr., Onfinite groups with an abelian Sylow group, Can. J. Math. 14 (1962), 436–450. Google Scholar

[9] 9. Brauer, R. and Tuan, H. F., On simple groups of finite order. I, Bull. Amer. Math. Soc. 51 (1945), 756–766. Google Scholar

[10] 10. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (J. Wiley and Sons, New York, 1962). Google Scholar

[11] 11. Feit, W., Groups which have a faithful representation of degree less than p-1, Trans. Amer. Math. Soc. 112 (1934), 287–303. Google Scholar

[12] 12. Feit, W. and Thompson, J. G., Groups which have a faithful representation of degree less than (p-l/2)(sic), Pacific J. Math. 11 (1961), 1257–1262. Google Scholar

[13] 13. Green, J. A., Blocks of modular representations, Math. Z. 79 (1962), 109–115. Google Scholar

[14] 14. Hall, M., The theory of groups (MacMillan, New York, 1959). Google Scholar

[15] 15. Hall, M. and Wales, D., The simple group of order 604800, J. Algebra 9 (1968), 417–450. Google Scholar

[16] 16. Lindsey, J. H. II, On a six-dimensional projective representation of the Hall-Janko group, Pacific J. Math. 35 (1970), 175–186. Google Scholar

[17] 17. Lindsey, J. H. II, On a six dimensional projective representation of PSU4(3), Pacific J. Math. 36 (1971), 407–425. Google Scholar

[18] 18. Lindsey, J. H. II, A generalization of Feit's theorem, Trans. Amer. Math. Soc. 155 (1971), 65–75. Google Scholar

[19] 19. Lindsey, J. H. II, Linear groups with an irreducible, normal, rank two p-subgroup (to appear). Google Scholar

[20] 20. Mitchell, H., Determination of all primitive collineation groups in more than four variables which contain homologies, Amer. J. Math. 36 (1914), 1–12. Google Scholar

[21] 21. Schur, I., Uber eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzber. Preuss. Akad. Wiss. (1905), 77–91. Google Scholar

[22] 22. Schur, I., Untersuchungen uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. fur Math. 132 (1907), 85–137. Google Scholar

[23] 23. Schur, I., Uber die Darstellung die symmetrischen lineare Substitutionen, J. fur Math. 139 (1911), 155–250. Google Scholar

[24] 24. Suzuki, M., Finite groups in which the centralizer of any element of order 2 is 2-closed, Ann. of Math. 82 (1965), 191–212. Google Scholar

[25] 25. Tuan, H. F., On groups whose orders contain a prime to the first power, Ann. of Math. 45 (1944), 110–140. Google Scholar

[26] 26. van der Waerden, B. L., Gruppen von linearen Transformationen, Ergebnisse der Mathematik und ihrer Grenzgebiete (Chelsea, New York). Google Scholar | DOI

[27] 27. Wales, D., Finite linear groups of prime degree, Can. J. Math. 21 (1969), 1025–1041. Google Scholar

[28] 28. Wales, D., Finite linear groups of degree seven. I, Can. J. Math. 21 (1969), 1042–1056. Google Scholar

[29] 29. Wales, D., Finite linear groups of degree seven. II, Pacific J. Math. 34 (1970), 207–235. Google Scholar

[30] 30. Zassenhaus, H., Neuer Beweis der Endlichkeit der Klassenzahl bel unimodularer Aquivalenz endlicher ganzzahliger Substitutions gruppen, Abhandl. math. Seminar Hamburg Univ. 12 (1938), 276–288. Google Scholar