Linear groups with orders having certain large prime divisors.(English)Zbl 1041.20035
Let \(q\) be a power of a prime \(p\). This substantial paper classifies the subgroups \(G\) of \(\text{GL}_d(q)\) with order divisible by a primitive prime divisor \(r\) of \(q^e-1\) for some \(e>d/2\) (that is, \(r\) divides \(q^e-1\) but does not divide \(q^i-1\) for \(1\leq i<e\)). This classification generalizes previous results ofC. Hering [Geom. Dedicata 2, 425-460 (1974;Zbl 0292.20045); J. Algebra 93, 151-164 (1985;Zbl 0583.20003)] andU. Dempwolff [Rend. Semin. Mat. Univ. Padova 77, 69-113 (1987;Zbl 0624.20008)].
One of the main ingredients of the authors’ approach is a fundamental theorem ofM. Aschbacher [Invent. Math. 76, 469-514 (1984;Zbl 0537.20023)], according to which \(G\) is either contained in a naturally defined subgroup of \(\text{GL}_d(q)\), or \(G\) is nearly simple (that is, \(S\leq G/Z(G)\leq\operatorname{Aut}(S)\) for some finite non-Abelian simple group \(S\)).
In the former case, the authors prove that \(G\) is one of the groups described in Examples 2.1-2.5 of the paper. The latter case leads to groups listed in Examples 2.6-2.9. In this case, the condition imposed on \(r\) implies that \(d\leq 2r-3\). Thus \(G\) has a faithful representation over \(\mathbb{F}_q\) of relatively small degree, and further analysis relies largely on knowledge of small representations of finite quasi-simple groups. For various \(S\) and to various extents, this information is available in the literature.
But in the cases where \(S\) is a (projective) special linear group, special unitary group, or symplectic group defined in characteristic other than \(p\), existing results are not strong enough for the authors’ purposes. So the authors prove another main result showing that, in these three cases, the faithful representations of \(G\) have degree either equal to one of a few prescribed values (and any of these values does occur in reality) or at least (roughly) twice of those values. This result improves previous results ofV. Landazuri andG. M. Seitz [J. Algebra 32, 418-443 (1974;Zbl 0325.20008)] and ofG. M. Seitz andA. E. Zalesskij [J. Algebra 158, No. 1, 233-243 (1993;Zbl 0789.20014)]. The main result of the paper has already been used to design Monte Carlo algorithms for recognizing finite classical groups and to study questions concerning generation of classical groups, and it will be widely used in many other applications.
One of the main ingredients of the authors’ approach is a fundamental theorem ofM. Aschbacher [Invent. Math. 76, 469-514 (1984;Zbl 0537.20023)], according to which \(G\) is either contained in a naturally defined subgroup of \(\text{GL}_d(q)\), or \(G\) is nearly simple (that is, \(S\leq G/Z(G)\leq\operatorname{Aut}(S)\) for some finite non-Abelian simple group \(S\)).
In the former case, the authors prove that \(G\) is one of the groups described in Examples 2.1-2.5 of the paper. The latter case leads to groups listed in Examples 2.6-2.9. In this case, the condition imposed on \(r\) implies that \(d\leq 2r-3\). Thus \(G\) has a faithful representation over \(\mathbb{F}_q\) of relatively small degree, and further analysis relies largely on knowledge of small representations of finite quasi-simple groups. For various \(S\) and to various extents, this information is available in the literature.
But in the cases where \(S\) is a (projective) special linear group, special unitary group, or symplectic group defined in characteristic other than \(p\), existing results are not strong enough for the authors’ purposes. So the authors prove another main result showing that, in these three cases, the faithful representations of \(G\) have degree either equal to one of a few prescribed values (and any of these values does occur in reality) or at least (roughly) twice of those values. This result improves previous results ofV. Landazuri andG. M. Seitz [J. Algebra 32, 418-443 (1974;Zbl 0325.20008)] and ofG. M. Seitz andA. E. Zalesskij [J. Algebra 158, No. 1, 233-243 (1993;Zbl 0789.20014)]. The main result of the paper has already been used to design Monte Carlo algorithms for recognizing finite classical groups and to study questions concerning generation of classical groups, and it will be widely used in many other applications.
Reviewer: Pham Huu Tiep (Gainesville)
MSC:
| 20G40 | Linear algebraic groups over finite fields |
| 20C33 | Representations of finite groups of Lie type |
| 20E28 | Maximal subgroups |
| 20E34 | General structure theorems for groups |
| 20G05 | Representation theory for linear algebraic groups |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
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