- ALGEBRAIC STRUCTURES 457
defines. Essentially, this representation was introduced by the American logician
philosopher Charles Sanders Peirce in 1879.
An early prefiguration of an important concept in the representation of groups
occurred in an 1837 paper of Dirichlet proving that an arithmetic progression whose
first term and difference are relatively prime contains infinitely many primes. As
discussed in Section 1 of Chapter 8, that paper contains the Dirichlet character
÷(ç), defined as (-l)(n-1)/^2 if n is odd and 0 if ç is even. This character has
the property that ÷{ôçç) = ÷{ôç)÷(ç). The definition of a character as a homo-
morphism into the multiplicative group of nonzero complex numbers was given by
Dedekind in an 1879 supplement to Dirichlet's lectures. About the same time,
Sylvester was showing how matrices could be used to represent the quaternions.^19
The theory of representations of finite groups was developed by the German
mathematician Ferdinand Georg Frobenius (1849-1917), responding to a question
posed by Dedekind. The original question was simply to factor the determinant of a
certain matrix associated with a finite group. In trying to solve this problem, Frobe-
nius introduced the idea of a representation and a character of a representation.
Although the subject is too technical for details to be given here, the characters,
being computable, reveal certain facts about the structure of a finite group, and in
some cases determine it completely.
This theory was extended to Lie groups in 1927 by Hermann Weyl and his
student F. Peter. In that context, it turned out, representation theory subsumes
and unifies the theories of Fourier series and Fourier integrals, both of which are
ways of analyzing functions defined on a Lie group (the circle or line) by transferring
them to functions defined on a separate (dual) group. The subject is now called
abstract harmonic analysis.
Finite groups. The subject of finite groups grew up in connection with the solution
of equations, as we have already seen. For that purpose, one of the most important
questions was to decide which groups corresponded to equations that are solvable.
Such a group G has a chain of normal subgroups G D Gi D G2 D ·•• D {1}
in which each factor group Gi/Gi+\ is commutative. Because of the connection
with equations, a group having such a chain of subgroups is said to be solvable.
A solvable group can be built up from the simplest type of group, the group of
integers modulo a prime, and so its structure may be regarded as known. It would
be desirable to have a classification that can be used to break down any finite group
into its simplest elements in a similar way. The general problem is so difficult that
it is nowhere near solution. However, a significant piece of the program has been
achieved: the classification of simple groups. A simple group is one whose only
normal subgroups are itself and {1}.
The project of classifying these groups was referred to by one of its leaders,
Daniel Gorenstein (1923-1992), as the Thirty Years' War, since a strategy for the
classification was suggested by Richard Brauer (1901-1977) at the International
Congress of Mathematicians in 1954 and the classification was completed in the(^19) Cayley, Peirce, and Sylvester were well acquainted personally and professionally with one an-
other. Cayley and Peirce were at Johns Hopkins University during part of the time that Sylvester
was chair of mathematics there. They formed the strong core of the Anglo-American school of
abstract algebra described by Parshall (1985). At the heart of this abstraction, at least in the
case of Peirce, was a philosophical program of creating a universal symbolic algebra that could be
applied in any situation. Such a program required that the symbols be mere symbols until applied
to some specific situation.