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Finite imprimitive linear groups of prime degree. (English) Zbl 1062.20051

The problem of classifying the finite groups which act irreducibly on a complex vector space of given dimension is an old one, which has been solved for small dimensions but is intractable in general. The paper under review is a contribution to the case when the dimension is prime. The problem splits naturally into the primitive and the imprimitive groups, and perhaps less naturally into the soluble and insoluble cases. The insoluble, imprimitive case is treated here, generalised to allow the underlying field to be algebraically closed of arbitrary characteristic.
Groups of this type are extensions of a (possibly trivial) Abelian group \(A\) by an insoluble transitive permutation group \(H\) of prime degree. Assuming the classification theorem for finite simple groups, \(H\) is 2-transitive and its socle is either \(A_r\) (with \(r\) prime) or \(\text{PSL}_n(q)\) (with \((q^n-1)/(q-1)\) prime; it is not known whether the number of such primes is finite or infinite), or \(\text{PSL}_2(11)\) or \(M_{11}\) or \(M_{23}\). The possible groups \(A\) are now determined using the modular representation theory of \(H\).
To classify the possible extensions \(A.H\) it is necessary to calculate some 2-cohomology: it is proved that the extension always splits unless there is a prime \(p\) such that both (a) the point stabilizer in \(H\) has a normal subgroup of index \(p\), and (b) \(p\) divides the order of \(A\). The possible extensions are not determined in this case. Finally, for each extension the irreducible faithful representations of the given degree need to be determined. Some examples of this are given.

MSC:

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20B20 Multiply transitive finite groups
20D05 Finite simple groups and their classification
20C15 Ordinary representations and characters

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