Why Beauty is Truth: A History of Symmetry by Ian Stewart
Another really good book by Ian Stewart. Like lots of other books around this area, it gives a basic introduction to group theory and discusses how Galois and others used this to show that there is no general solution to the quintic. As well as providing historical data about the characters involved, this book does a really good job at describing the mathematics behind the proof in a really easy to understand way. Better still, rather than finishing at this point, the book then goes on to discuss Lie groups and their relationship to relativity and quantum mechanics.