Chapter 2

Groups

Wherever groups disclose themselves, or could be introduced, simplicity crystallizes out of complete chaos.

—Eric Temple Bell

The origin of the modern theory of groups lies in the theory of equations. By the beginning of the nineteenth century, mathematicians had developed formulas for finding the roots of any cubic or quartic equation (analogous to the quadratic formula), and the best mathematicians of the day were trying to find such a formula for the quintic. It thus came as a great surprise when, in 1824, Niels Henrik Abel proved that no such formula exists. At about the same time, Evariste Galois showed that any equation of degree n has an associated group of permutations of the roots of the equation (that is, a set of permutations closed under compositions and inverses). He proved that the equation is solvable if and only if this group has a certain property (now called a solvable group). In particular, the fact that the group A_{n} of even permutations is not solvable for any n ≥ 5 implies that no formula exists for solving equations of degree n ≥ 5. This spectacular achievement led to modern Galois theory, but Galois' work went unrecognized until after his death at age 20.

Galois worked with groups of permutations. Then, in 1854, Arthur Cayley formulated the abstract group concept. While the study of permutation groups continues to occupy mathematicians, the abstract theory has the advantage that it isolates those properties of groups that do not depend ...