×

On the composition length of finite primitive linear groups. (English) Zbl 1015.20034

An irreducible subgroup \(G\) of \(\text{GL}_K(V)\) is called quasi-primitive if every normal subgroup of \(G\) is homogeneous on \(V\). Let \(K\) be a finite field or a number field, \(V\) a \(K\)-vector space, and \(G\) a finite quasi-primitive subgroup of \(\text{GL}_K(V)\). The authors bound the composition length of \(G\) in terms of the dimension of \(V\) over \(K\) and the degree of \(K\) over its prime subfield. As a by-product, the authors prove a result of number theory which bounds the number of prime factors (counting multiplicities) of \(q^n-1\) (\(q,n\geq 1\) integers), improving a result of A. Turull and A. Zame [Arch. Math. 55, No. 4, 333-341 (1990; Zbl 0691.10035)].
Reviewer: Li Fu-an (Beijing)

MSC:

20G40 Linear algebraic groups over finite fields
20G30 Linear algebraic groups over global fields and their integers
11N56 Rate of growth of arithmetic functions
11N05 Distribution of primes
20D30 Series and lattices of subgroups

Citations:

Zbl 0691.10035
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. Apostol, Introduction to Analytic Number Theory. Berlin-Heidelberg-New York 1976. · Zbl 0335.10001
[2] M. Aschbacher, Finite Group Theory. Cambridge 1986. · Zbl 0583.20001
[3] B. Huppert, Endliche Gruppen I. Berlin-New York 1967. · Zbl 0217.07201
[4] A. Lucchini, F. Menegazzo andM. Morigi, On the number of generators and composition length of finite linear groups. J. Algebra243, 427–447 (2002). · Zbl 0992.20033 · doi:10.1006/jabr.2001.8882
[5] J. J. Sylvester, On the divisors of the sum of a geometrical series whose first term is unity and common ratio any positive or negative integer. Nature37, 417–418 (1888) and Collected Papers, vol. IV, 625–629, Cambridge 1912. · JFM 20.0176.05 · doi:10.1038/037417a0
[6] A. Turull andA. Zame, Number of prime divisors and subgroup chains. Arch. Math.55, 333–341 (1990). · Zbl 0691.10035 · doi:10.1007/BF01198471
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.