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On linear groups of degrees at most \(|{} P|{} -1\). (English) Zbl 0765.20019

The structure of finite complex linear groups \(G\) of degree at most \(| P|-1\) with a T. I. Sylow \(p\)-subgroup \(P\) is determined. If \(\overline {G}=G/Z(G)\) is nonabelian simple and \(\chi\) is a faithful complex character of \(G\) of degree at most \(| P|-1\), then one of the following holds:
1) \(\overline{G}=SL_ 2(2^ n)\), \(| P|=p=1+2^ n\), \(\chi(1)\geq p-2\); 2) \(\overline{G}=PSL_ 3(4)\), \(\chi\) is irreducible of degree 8; 3) \(P\) is cyclic, \(\chi\) is irreducible of degree \(| P|-1\), \(\overline{G}\) is isomorphic to \(PSL_ n(q)\) with \(n\geq 3\) and \(| P|=(q^ n-1)/(q-1)\); \(PSU_ n(q)\) with \(n\) odd and \(| P|=(q^ n+1)/(q-1)\); \(PSp_{2n}(q)\) with \(| P|=(q^ n+1)/2\); \(M_{11}\), \(M_{12}\) or \(M_{22}\) with \(| P|=11\); \(G_ 2(4)\) or \(Suz\) with \(| P|=13\); \(J_ 3\) with \(| P|=19\); \(M_{23}\) with \(| P|=23\); \(Ru\) with \(| P|=29\); or \(A_ p\); 4) \(\overline{G}\) is isomorphic to \(PSL_ 2(q)\) with \(| P|=(q+1)/(2,q-1)\), \(q\) or \(q+1\) and \(\chi(1)\geq(q- 1)/(2,q-1)\); \(PSU_ 3(q)\) with \(| P|=q^ 3\), \(q>2\) and \(\chi(1)\geq q(q-1)\); \(^ 2G_ 2(q)\) with \(| P|=q^ 3\) and \(\chi(1)\geq q^ 2-q+1\); \(^ 2B_ 2(q)\) with \(| P|=q^ 2\), \(q=2^{2m+1}\) and \(\chi(1)\geq \sqrt{q/2}(q-1)\); or \(Mc\) with \(| P|=125\) and \(\chi(1)\geq 22\).
This generalizes and extends the results of H. Blau [J. Algebra 114, 268-285 (1988; Zbl 0638.20014)] and the author [Contemp. Math. 82, 243-254 (1989; Zbl 0671.20032)]. The classification of finite simple groups and estimates from the work of V. Landazuri and G. Seitz [J. Algebra 32, 418-443 (1974; Zbl 0325.20008)] and the Atlas of Finite Groups [J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson (Clarendon, Oxford 1985; Zbl 0568.20001)] are used heavily throughout the article.

MSC:

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20C15 Ordinary representations and characters
20D05 Finite simple groups and their classification
20G05 Representation theory for linear algebraic groups
20H20 Other matrix groups over fields
20C34 Representations of sporadic groups
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References:

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