= Solution
{c}
<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.<Beweis des Satzes über Primzahlen und quadratische Formen by Heinrich Weber>[Mathematische Annalen 20 (1882), 301–329; opening group definition as reproduced in the exam]{ref}<The abstract group concept by the MacTutor History of Mathematics Archive>[Discussion of Weber's 1882 definition and its class-group setting]{ref}
The apparent omission of an <identity element> and <inverse elements> is not fatal. \b[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 $x$,
$$
a(ex)=(ae)x=ax,
$$
and left cancellation gives $ex=x$. To show $e$ is also a right <identity element>, use
$$
(xe)x=x(ex)=xx.
$$
Right 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>,
$$
c=ce=c(ab)=(ca)b=eb=b.
$$
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 \b[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.
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