Full Description
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.
Contents
Background theory.- The imprimitive soluble subgroups of GL(2, p k ).- The normaliser of a Singer cycle of prime degree.- The irreducible soluble subgroups of GL(2, p k ).- Some irreducible soluble subgroups of GL(q, p k ), q>2.- The imprimitive soluble subgroups of GL(4, 2) and GL(4, 3).- The primitive soluble subgroups of GL(4, p k).- The irreducible soluble subgroups of GL(6, 2).- Conclusion.- The primitive soluble permutation groups of degree less than 256.
