
Chapter 14
Solubility and Simplicity
In order to apply the Galois correspondence, in particular to solving equations by
radicals, we need to have at our fingertips a number of group-theoretic concepts and
theorems. We have already assumed familiarity with elementary group theory: sub-
groups, normal subgroups, quotient groups, conjugates, permutations (up to cycle
decomposition): to these we now add the standard isomorphism theorems. The rel-
evant theory, along with most of the material in this chapter, can be found in any
basic textbook on group theory, for example Fraleigh (1989), Humphreys (1996), or
Neumann, Stoy, and Thompson (1994).
We start by ...