The Primitive Soluble Permutation Groups of Degree Less than 256

1. Front Matter

   Pages I-IX

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2. Introduction

   • Mark W. Short

   Pages 1-9

3. Background theory

   • Mark W. Short

   Pages 10-42

4. The imprimitive soluble subgroups of GL(2, p ^k)

   • Mark W. Short

   Pages 43-54

5. The normaliser of a Singer cycle of prime degree

   • Mark W. Short

   Pages 55-61

6. The irreducible soluble subgroups of GL(2, p ^k)

   • Mark W. Short

   Pages 62-74

7. Some irreducible soluble subgroups of GL(q, p ^k), q>2

   • Mark W. Short

   Pages 75-83

8. The imprimitive soluble subgroups of GL(4, 2) and GL(4, 3)

   • Mark W. Short

   Pages 84-92

9. The primitive soluble subgroups of GL(4, p ^k)

   • Mark W. Short

   Pages 93-107

10. The irreducible soluble subgroups of GL(6, 2)

    • Mark W. Short

    Pages 108-113

11. Conclusion

    • Mark W. Short

    Pages 114-120

12. Erratum

    • Mark W. Short

    Pages 146-146

13. Back Matter

    Pages 121-145

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This monograph addresses the problem of describing all primitive soluble permutation groups of a given degree, with particular reference to those degrees less than 256. The theory is presented in detail and in a new way using modern terminology. A description is obtained for the primitive soluble permutation groups of prime-squared degree and a partial description obtained for prime-cubed degree. These descriptions are easily converted to algorithms for enumerating appropriate representatives of the groups. The descriptions for degrees 34 (die vier hochgestellt, Sonderzeichen) and 26 (die sechs hochgestellt, Sonderzeichen) are obtained partly by theory and partly by machine, using the software system Cayley. The material is appropriate for people interested in soluble groups who also have some familiarity with the basic techniques of representation theory. This work complements the substantial work already done on insoluble primitive permutation groups.

• Finite Groups
• Group Theory
• Permutation
• Prime
• Representation theory

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