Past exam of the history of mathematics course of the University of Oxford / 2023 / 6

Heinrich Weber's 1882 paper concerns prime numbers represented by quadratic forms. Its definition abstracts a finite group from the nature of its elements: what matters is a closed binary operation, associativity, and left and right cancellation laws. Here degree $h$ means the number of elements, now called the group order; it is not the number of letters acted on by a permutation group. This matters when relating the passage to earlier work on substitutions. The definition fits the use of groups of classes of quadratic forms, where the elements are already mathematical objects more elaborate than individual permutations.^[ref][ref]

The apparent omission of an identity element and inverse elements is not fatal. Every nonempty finite cancellative semigroup is a group. Here is a proof identifying exactly where the assumption of a finite set enters. For fixed $a\in G$, the maps $L_a(x)=ax$ and $R_a(x)=xa$ are injective by cancellation. An injective function from a finite set to itself is surjective, so choose $e$ with $ae=a$. For every $x$,and left cancellation gives $ex=x$. To show $e$ is also a right identity element, useRight cancellation gives $xe=x$. Thus the same $e$ works on both sides for every $x$; it has not merely been chosen as an element fixing $a$.

Surjectivity now gives $b,c$ with $ab=e$ and $ca=e$. By associativity,So $b$ is a two-sided inverse element of $a$. Since $a$ was arbitrary, the usual group axioms follow. The nonempty hypothesis is tacit in a system of $h$ elements intended to have positive $h$: the empty system would satisfy the displayed universal conditions vacuously but would have no identity element.

The infinite case exposes the force of the qualification. The positive integers under addition are closed, associative and cancellative, but have no additive identity element. Even the nonnegative integers have an identity element without additive inverse elements for positive elements. The failure is precisely that injective functions of an infinite set to itself need not be surjective. Heinrich Weber's conditions are therefore not an unrestricted replacement for the modern group axioms.

Historically the abstraction is important because it allows arguments to be applied across different kinds of elements. An assumption of commutativity is absent: class-group applications do not make commutativity part of the general definition. For example, the symmetric group $S_3$ meets the conditions but is noncommutative. The definition also does not require a prior presentation as permutations, although Cayley's theorem provides such a representation. The passage documents the separation of an algebraic structure from any one representation of it, with finiteness carrying logical work that later axiom systems usually assign to explicit identity element and inverse element axioms.