The primitive soluble permutation groups of degree less than 256
These notes are motivated by an algorithm of \textit{L. G. Kovacs, J. Neubüser, M. F. Newman} [``Some algorithms for finite soluble groups'', unpublished notes] for computing certain maximal subgroups of low index. Investigated are the finite primitive soluble permutation groups, or, equivalently, the irreducible soluble subgroups of \(\mathrm{GL}(n,p)\). The author has two main objectives. One of these objectives is to develop efficient algorithms that take as input a positive integer \(n\) and a prime \(p\), and produce a list of the irreducible soluble subgroups of \(\mathrm{GL}(n,p)\). The other main objective is to execute these algorithms for those \(n\) and \(p\) such that \(p^ n< 256\), and to provide electronic access to the list of groups so obtained. A secondary objective is to ``match'', as far as possible, the list of \textit{J. D. Dixon} and \textit{B. Mortimer} [Math. Proc. Camb. Philos. Soc. 103, No. 2, 213--238 (1988; Zbl 0646.20003)] of all finite primitive permutation groups with insoluble socles whose degree is less than 1000. The final chapter concludes with a note on work in progress on the irreducible soluble subgroups of \(\mathrm{GL}(8,2)\).
finite primitive soluble permutation groups
irreducible soluble subgroups of \(GL(n,p)\)
scientific article; zbMATH DE number 53711
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