Simple Groups

We saw in Chapter 19 that the group concept came to light when Galois used it to explain why some equations are solvable and some are not. Solving an equation corresponds to “simplifying” a group by forming quotients, so knowing which equations are

not

solvable depends on knowing which groups cannot be “simplified.” These are the so-called

simple

groups. The groups associated with polynomial equations are finite, so one would like to classify the finite simple groups. Galois found one infinite family of such groups—the alternating groups

An

for

n

≥ 5—and three other provocative examples that we now view as the symmetry groups of finite projective lines. However, classification of the finite simple groups was much harder than could have been foreseen in the 19th century. It turned out to be easier (though still very hard) to classify

continuous

simple groups. This was done by Lie, Killing, and Cartan in the 1880s and 1890s. Each continuous simple group is the symmetry group of a space with hypercomplex coordinates, either from ℝ, ℂ,ℍ, or

$$\mathbb{O}$$

. While this classification was in progress, it was noticed that a single continuous simple group can yield infinitely many finite simple groups, obtained by replacing the hypercomplex number system by a finite field. These “finite groups of Lie type” were completely worked out by 1960. Together with the alternating groups and the cyclic groups of prime order, they account for all but finitely many of the finite simple groups. But identifying all the exceptions—the 26

sporadic

simple groups— turned out to be the hardest problem of all…