Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

understanding proofs 341the use of the concept of a group to establish a proposition in elementarynumber theory.A group consists of a set, G, an associative binary operation, ·, onG, andan identity element, 1, satisfying 1 · a = a · 1 = a for every a in G. In agroup, every element a is assumed to have an inverse, a −1 , satisfying a · a −1 =a −1 · a = 1. It is common to use the letter G to refer to both the groupand the underlying set of elements, even though this notation is ambiguous.We will also write ab instead of a · b. Exponentiation a n is defined to be thethe product a · a ···a of n copies of a, andifS is any finite set, |S| denotes thenumber of elements in S.Proposition 1. Let G be a finite group and let a be any element of G. Then a |G| = 1.This proposition is an easy consequence of a theorem known as Lagrange’stheorem, but there is an even shorter and more direct proof when the group isabelian, that is, when the operation satisfies ab = ba for every a and b.Proof of Proposition 1 when G is abelian. GivenG and a, consider the operationf a (b) = ab, that is, multiplication by a on the left. This operation is injective, sinceab = ab ′ implies b = a −1 ab = a −1 ab ′ = b ′ ,andsurjective,sinceforanyb we canwrite b = a(a −1 b). Sof a is a bijection from G to G.Suppose the elements of G are b 1 , ..., b n . By the preceding paragraph, multiplicationby a simply permutes these elements, so ab 1 , ..., ab n also enumerates theelements of G. The product of these elements is therefore equal to both(ab 1 )(ab 2 ) ···(ab n ) = a n (b 1 b 2 ···b n ),and b 1 b 2 ···b n .Thisimpliesa n = 1, as required.□To apply Proposition 1 to number theory, we need only find a suitablegroup. Write a ≡ b(m), and say that a and b are congruent modulo m, ifaand b leave the same remainder on division by m, that is, if m dividesa − b. The relationship of being congruent modulo m is symmetric, reflexive,and transitive. It also respects addition and multiplication: if a ≡ a ′ (m) andb ≡ b ′ (m), then(a + a ′ ) ≡ (b + b ′ )(m) and aa ′ ≡ bb ′ (m). Anyinteger,a, iscongruent to a number between 0 and m − 1 modulo m, namely, its remainder,or ‘residue’, modulo m. Thus congruence modulo m divides all the integersinto m ‘equivalence classes’, each represented by the corresponding residue.We can think of addition and multiplication as operations on these equivalenceclasses, or, alternatively, as operations on residues, where after each operationwe take the remainder modulo m. (For example, under clock arithmetic, weonly care about the values 0, ... , 11, and we are adding modulo 12 when wesay that ‘5 hours after 9 o’clock, it will be 2 o’clock’.)