(^458) 15. MODERN ALGEBRA
1980s with the discovery of the last "sporadic" group. The project consists of so
many complicated parts that the process of streamlining and clarifying it, with all
the new projects that process will no doubt spawn, is likely to continue for many
decades to come.
An important part of the project was the 1963 proof by Walter Feit and John
Thompson (b. 1932) that all finite simple groups have an even number of elements.
The proof was 250 pages long and occupied an entire issue of the Pacific Journal of
Mathematics. As it turned out, this project was destined to generate large numbers
of long papers. In fact, the mathematician who contributed the last step in the
classification later wrote, "At least 3,000 pages of mathematically dense preprints
appeared in the years 1976-1980 and simply overwhelmed the digestive system of
the group theory community" (Solomon, 1995, p. 236). The outcome of the project
is intensely satisfying from an aesthetic point of view. It turns out that there are
three (or four) infinite classes of finite simple groups: (1) groups of prime order;
(2) the group of even permutations on ç letters (ç > 5);^20 (3) certain finite linear
groups, a class that can be subdivided into classical matrix groups and twisted
groups of Lie type, whose exact definition is not important for present purposes.
Outside those classes are the "sporadic" groups.
If this classification seems to resemble the old classification of constructions as
planar, solid, and curvilinear, in which the final class is merely a catchall term for
anything that doesn't fit into the other classes, that impression is misleading. The
class of sporadic groups turns out to contain precisely 26 groups, whose properties
have been tabulated. The smallest of them is Ë/ð with 7920 elements, one of
five sporadic simple groups discovered by Emil Mathieu (1835 1890). The largest,
officially denoted F\, is informally known as the Monster, since it consists of
2 46.^320. 59. 7 e. Ð^2. ]33. 17. 19. 23 •^29 •^31 •^41 • 47 · 59 · 71
elements. It was constructed in 1980.
2.3. Number systems. Rings and fields can be regarded as generalized number
systems, since they admit addition, subtraction, and multiplication, and sometimes
division as well. As noted above, such general systems have been used since Gauss
began the study of arithmetic modulo an integer and proved that the factorization
of a Gaussian integer m + ni into irreducible Gaussian integers is unique up to a
power of i. Gauss was particularly interested in these numbers, since when they
are introduced, 2 is not a prime number (2 = (1 + i)(l — i)), nor is any number of
the form 4n + 1 prime. For example. 5 = (2 + i)(2 — i). The number 3, however,
remains prime. If we pass to numbers of the form m + n\/—2, factorization is
still unique, but this uniqueness is lost for numbers of the form m 4- ni/-?, since
4 = 2- 2 = (1 + - >/~3)- The Gaussian integers were the first of an
increasingly abstract class of structures on which multiplication is defined and obeys
a cancellation law (that is ac = be and c ö 0 implies that á = 6), but division is
not necessarily always possible. If the factorization of each element is essentially
unique, such a structure is called a Gaussian domain.
(^20) The fact that this group is simple and noncommutative implies that the symmetric group
consisting of all permutations on ç letters cannot be solvable for ç > 5, and hence that the
general equation of degree ç is not solvable by radicals.
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