paper Finite primitive linear groups of prime degree
نویسنده
چکیده
In our paper referred to above we claim to enumerate all nite primitive linear groups of prime degree r over C with a nonabelian socle. However, the case where the socle is imprimitive was overlooked. In the present paper we deal with this case to complete the classi cation. 2000 Mathematics Subject Classi cation: 20H20 20C15 20C33 In the paper referred to above, we state a theorem (Theorem 1.2) in which we claim to enumerate all nite primitive subgroups G of SL(r;C) with r prime for which G=Z(G) has a nonabelian socle M=Z(G). We are indebted to Professor Ziping Zhang (Peking University, Beijing) for pointing out that our classi cation fails to include the cases where G is primitive but M is imprimitive. In the present note we deal with this latter case. Theorem 1 Let r be prime. Suppose that there exists a nite primitive group G GL(r;C) such that the socle M=Z(G) of G=Z(G) is nonabelian and M is imprimitive. Then the derived group G0 is imprimitive and for some n and q we have G0 = PSL(n; q) and r = (qn 1)=(q 1) 5; moreover, if n = 2 then q is even, and if n > 2 then q is odd except in when (n; q) = (3; 2). Conversely, given any integer n and prime power q satisfying these side conditions, if r = (qn 1)=(q 1) 5 then there exists a nite primitive group G GL(r;C) such that G0 is imprimitive and isomorphic to PSL(n; q). (In the latter case the socle of G=Z(G) contains PSL(n; q) as a composition factor and so is nonabelian.) Futher information, including other arithmetic restrictions on n and q and restrictions on the possible representations of PSL(n; q), can be deduced from the lemmas below. For example, n must always be a prime and r is a
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