The History of Mathematics: A Brief Course

454 15. MODERN ALGEBRA

the structure constants that constitute the multiplication table for the elements of

a basis. Even so, the subject seems to have caught on in only a few places in

Germany. At Gottingen Emmmy Noether revolutionized the subject of algebras

and representations of finite groups, and the concept of a Noetherian ring is now

one of the basic parts of ring theory. Yet Salomon Bochner (1899-1982), who was

educated in Germany and spent the first 15 years of his professional career there

before coming to Princeton, recalled that the concept of an algebra was completely

new to him when he first heard a young American woman lecture on it at Oxford

in 1925.^18

She came from Chicago and she gave at this seminar a lecture on

algebras, which left all of us totally uncomprehending what it was

all about. She spoke in a well-articulated, self-confident manner,

but none of us had remotely heard before the terms she used, and

we were lost. [Bochner, 1974, p. 832]

Bochner went on to say that a decade later he was taken aback to find a German

book with the title Algebren (Algebras- it was use of the plural that Bochner found

jarring). Bochner was an extremely creative and productive analyst and differential

geometer. Not until the mid-1960s did he have time to ferret out Peirce's book and

sit down to read it.

2.2. Abstract groups. The general theory of groups of permutations was devel-

oped in great detail in an 1869 treatise of Camille Jordan (1838-1922). This work

made the importance of groups widely known. But despite Cayley's 1849 paper in

group theory, the word group was still being used in an imprecise sense as late as

1871, in the work of two of the founders of group theory, Felix Klein and Sophus

Lie (see Hawkins, 1989, p. 286). All groups were pictured concretely, as one-to-one

mappings of sets. That assumption made a cancellation law ab = ac => b — c

valid automatically. For permutations of finite sets, the cancellation law implied

that every clement had an inverse. The corresponding inference for mappings of

infinite sets is not valid, but Klein and Lie did not notice the difference at first.

Lie even thought this inference could be proved. Groups as an abstract concept,

characterized by the three, four, or five axioms one finds in modern textbooks, did

not arrive until the twentieth century. On the abstract concept of a group Klein

(1926, pp. 335-336) commented:

This abstract formulation is helpful in the construction of proofs,

but not at all adapted to the discovery of new ideas and methods;

on the contrary, it rather represents the culmination of an earlier

development.

Klein was quick to recognize the potential of groups of transformations as a
useful tool in the study of many areas of geometry and analysis. The double peri-
odicity of elliptic functions, for example, meant that these functions were invariant

(^18) The woman's name was Echo Dolores Pepper, and the Mathematics Genealogy website lists
her as having received the Ph. D. at the University of Chicago in 1925. Her dissertation, Theory
of Algebras over a Quasi-field is on record, and she published at least one paper the year after
receiving the degree, "Asymptotic expression for the probability of trials connected in a chain,"
Annals of Mathematics, 2 (28) (1926 27), 318 326. I have been unable to find out any more
about her.