Generating finite completely reducible linear groups
HTML articles powered by AMS MathViewer
- by L. G. Kovács and Geoffrey R. Robinson PDF
- Proc. Amer. Math. Soc. 112 (1991), 357-364 Request permission
Abstract:
It is proved here that each finite completely reducible linear group of dimension $d$ (over an arbitrary field) can be generated by $\left \lfloor {\frac {3}{2}d} \right \rfloor$ elements. If a finite linear group $G$ of dimension $d$ is not completely reducible, then its characteristic is a prime, $p$ say, and the factor group of $G$ modulo the largest normal $p$-subgroup ${\mathbb {O}_p}\left ( G \right )$ may be viewed as a completely reducible linear group acting on the direct sum of the composition factors of the natural module for $G$: consequently, $G/{\mathbb {O}_p}\left ( G \right )$ can still be generated by $\left \lfloor {\frac {3}{2}d} \right \rfloor$ elements.References
- David M. Bloom, The subgroups of $\textrm {PSL}(3,\,q)$ for odd $q$, Trans. Amer. Math. Soc. 127 (1967), 150–178. MR 214671, DOI 10.1090/S0002-9947-1967-0214671-1
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- R. K. Fisher, The number of generators of finite linear groups, Bull. London Math. Soc. 6 (1974), 10–12. MR 352270, DOI 10.1112/blms/6.1.10
- Wolfgang Gaschütz, Zu einem von B. H. und H. Neumann gestellten Problem, Math. Nachr. 14 (1955), 249–252 (1956) (German). MR 83993, DOI 10.1002/mana.19550140406
- Robert M. Guralnick, On the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), no. 6, 521–523. MR 1023965, DOI 10.1007/BF01199809
- R. W. Hartley, Determination of the ternary collineation groups whose coefficients lie in the $\textrm {GF}(2^n)$, Ann. of Math. (2) 27 (1925), no. 2, 140–158. MR 1502720, DOI 10.2307/1967970
- Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
- I. M. Isaacs, The number of generators of a linear $p$-group, Canadian J. Math. 24 (1972), 851–858. MR 310053, DOI 10.4153/CJM-1972-084-0
- L. G. Kovács, On tensor induction of group representations, J. Austral. Math. Soc. Ser. A 49 (1990), no. 3, 486–501. MR 1074515, DOI 10.1017/S1446788700032456
- L. G. Kovács and Cheryl E. Praeger, Finite permutation groups with large abelian quotients, Pacific J. Math. 136 (1989), no. 2, 283–292. MR 978615, DOI 10.2140/pjm.1989.136.283
- Andrea Lucchini, A bound on the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), no. 4, 313–317. MR 1015993, DOI 10.1007/BF01195209
- James Wiegold, Growth sequences of finite groups, J. Austral. Math. Soc. 17 (1974), 133–141. Collection of articles dedicated to the memory of Hanna Neumann, VI. MR 0349841, DOI 10.1017/S1446788700016712
- James Wiegold, Growth sequences of finite groups. III, J. Austral. Math. Soc. Ser. A 25 (1978), no. 2, 142–144. MR 499355, DOI 10.1017/S1446788700038726
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 112 (1991), 357-364
- MSC: Primary 20H20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1047004-0
- MathSciNet review: 1047004