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

该项目属于EPSRC代数研究领域，更具体地说，属于群论研究领域，与计算群论密切相关并受其启发。在过去的50年里，许多有效的算法被设计，分析和实现，用于有限置换群（即有限对称群Sym（n）的子群）的计算。由于商群构成了群论的基本组成部分，因此自然会问，对于Sym（n）的子群G，最小的m是多少，使得G的商群G/N作为Sym（m）的子群出现。不幸的是，m通常可以指数地大于n。另一方面，许多特定的例子已经确定，其中m是最多n -这发生在例如当N是G的最大正规可解子群时。在这个项目中，学生将试图找到进一步的条件下，m是最多n，或者，不那么雄心勃勃，当m可以由n的多项式函数的边界。在另一个方向上也有一些有趣的例子，其中N很小，G/N有一个小度置换表示，但G没有。一个例子是当G = Alt（n）或Sym（n）时，|N| = 2，其中N在[G，G]中。在这种情况下，确定G的最小度置换表示仍然是有趣的，如果只是为了找出事情会变得多么糟糕，任何新的结果沿着这些路线都可能导致有限群计算算法的改进，反过来，通过广泛使用的软件包，如Sage，GAP和Magma，在离散数学计算的各个方面都有应用。

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.