Bounding the degree of permuation representations of quotient groups of finite permuation groups

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This project is in the EPSRC Research Area of Algebra and, more specifically, in Group Theory, and closely related to and motivated by Computational Group Theory.Over the past 50 years, many efficient algorithms have been devised, analysed and implemented for computing with finite permutation groups (i.e. subgroups of finite symmetric groups Sym(n)). Since quotient groups form a fundamental component of group theory, it is natural to ask, for a subgroup G of Sym(n), what is the smallest m such that a quotient group G/N of G arises as a subgroup of Sym(m). Unfortunately, m can be exponentially larger then n in general. On the other hand, many specific instances have been identified in which m is at most n - this occurs for example when N is the largest normal solvable subgroup of G.In this project, the student will attempt to find further conditions under which m is at most n or, less ambitiously, when m can be bounded by a polynomial function of n. One such condition that is worth investigating is when the composition factors of N are all nonabelian.There are also some interesting examples in the other direction, where N is very small, G/N has a small degree permutation representation, but G does not. An example of this is when G = Alt(n) or Sym(n) and |N| = 2 with N in [G,G]. It will be still be interesting to determine the smallest degree permutation representation of G in such cases, if only to find out just how bad things can get.Any new results along these lines are likely to result in improvements to algorithms for computing in finite groups which, in turn, via widely used software packages, such as Sage, GAP, and Magma, have applications to all aspects of computing in discrete mathematics.