Sylow subgroups

Abstract

This chapter has demonstrated how homomorphisms can be used to reduce the problem of computing a Sylow p-subgroup to smaller cases. This divide- and-conquer approach to analysing permutation groups and solving problems is very general and powerful. We will see further examples in later chapters.

The reductions mentioned in this chapter can be combined with each other and combined with the cyclic extension method or methods for treating soluble permutation groups. The best mix is a matter of engineering and is still the focus of further investigation.