Solvable Groups, Double Cosets and Isomorphism Theorems

We shall consider in this chapter a number of concepts which play an extremely important role in algebra in general. Here, for the most part, the theorems established will be introductory, and will subsequently be used to establish much deeper theorems. We shall begin with a discussion of a special class of groups called solvable groups. Later, we shall return to such groups, and we shall then give an alternate characterization of them and prove more properties related to solvable groups. This name comes from the fact that solvable groups are used in a subject called Galois Theory (also a part of algebra but not treated here) to determine whether or not a polynomial equation is solvable in terms of taking $n-th$ roots; i.e., to determine whether or not a formula for the roots of a polynomial like the quadratic formula (case $n =2$) can be found. If such a formula can be found, we say the polynomial equation is solvable by radicals. It turns out that not all polynomial equations of degree $\geq 5$ are solvable are by radicals. The reason for this is that the symmetric group Sn is not a solvable group for $n \geq 5$ (cf. Theorem 6.3.2).

We also discuss in the final section the so called correspondence theorem and two of the three isomorphism theorems. These theorems describe relationships between factor groups, normal subgroups, and homomorphisms. The reader should be cautioned that neither the numbering nor the content of these theorems is standard. So, for example, what is called the second isomorphism theorem here may be called the third isomorphism theorem in another text. Also some authors include what we have called the Fundamental Homomorphism Theorem (Theorem 7.1.8) as one of the isomorphism theorems. It should also be pointed out that analogs of these theorems are true for almost every type of algebraic system. (Examples where these hold other than groups, which the reader may be familiar with, are vector spaces.)