Linear groups of small degree over fields of finite characteristic
Finite linear groups over a field \(F\) of characteristic \(p>0\) which have degree less than \(p\) and are generated by their \(p\)-elements are classified. For each of these groups it is pointed out whether it can be lifted to characteristic 0. The whole list is too large to be given here. We shall state the answer for irreducible quasisimple groups only. Let \(G\) be a finite group, \(S=G/Z(G)\) be simple, \(V\) be a faithful irreducible \(G\)-module over \(F\) and \(\dim V<p\). Then one of the following holds: 1) \(\dim V=p-1\) and the pair \((S,p)\) appears in List 1; 2) \(\dim V=p-2\) and the pair \((S,p)\) appears in List 2; 3) \(p=7\), \(S=A_7\), \(\dim V=4\); 4) \(p=11\), \(S=J_1\), \(\dim V=7\); 5) \(S\) is a group of Lie type defined over a field of characteristic \(p\). List 1, 1) \(S=\text{PSp}(2n,q)\), \(n=2^a\), \(a\geq 0\), \(q\) is odd, \(p=(q^n+1)/2\); 2) \(S=\text{PSU}(n,q^2)\) (\(n\) is an odd prime, \(n\nmid(q+1)\)), \(p=(q^n+1)/(q+1)\); 3) \(S=A_5\), \(p=3\); 4) \(S=A_7\), \(p=5\); 5) \(S=\text{PSL}(3,4)\), \(\text{PSU}(4,9)\), \(J_2\), or \(A_7\), \(p=7\); 6) \(S=M_{11}\), \(M_{12}\), or \(M_{22}\), \(p=11\); 7) \(S=Suz\) or \(G_2(4)\), \(p=13\); 8) \(S=J_3\), \(p=19\); 9) \(S=Ru\), \(p=29\). List 2, 1) \(S=A_n\), \(n\) is prime, \(n\geq7\), \(p=n\); 2) \(S=\text{PSL}(n,q)\), \(n\) is prime, \(n\nmid(q-1)\), \(p=(q^n-1)/(q-1)\); 3) \(S=A_6\) or \(A_7\), \(p=5\); 4) \(S=M_{11}\), \(p=11\); 5) \(S=M_{23}\), \(p=23\). For the complex field the problem was solved by the second author [in: Contemp. Math. 82, 243-254 (1989; Zbl 0671.20032)]. The first author [J. Algebra 131, No. 1, 92-109 (1990; Zbl 0696.20013)] classified the linear groups in characteristic \(p\) with a cyclic \(p\)-Sylow subgroup \(P\) and degree \(d\leq|P|-2\), the second one [Proc. Am. Math. Soc. 110, No. 1, 53-57 (1990; Zbl 0724.20031)] did the same for \(d<p-2\), under the assumption that \(P\) is Abelian. The classification of finite simple groups and results of the first author [op. cit.] are used heavily in the article under review.
scientific article; zbMATH DE number 571351