APP. B] ALGEBRAIC SYSTEMS 439
EXAMPLE B.10
(a) The nonzero rational numbersQ\{0} form an abelian group under multiplication. The number 1 is the identity
element andq/pis the multiplicative inverse of the rational numberp/q.(b) LetSbe the set of 2×2 matrices with rational entries under the operation of matrix multiplication. Then
Sis not a group since inverses do not always exist. However, letGbe the subset of 2×2 matrices with a
nonzero determinant. ThenGis a group under matrix multiplication. The identity element isI=[
10
01]
and the inverse ofA=[
ab
cd]
isA−^1 =[
d/|A| −b/|A|
−c/|A| a/|A|]This is an example of a nonabelian group since matrix multiplication is noncommutative.(c) Recall thatZmdenotes the integers modulom. Zmis a group under addition, but it is not a group under
multiplication. However, letUmdenote a reduced residue system modulomwhich consists of those integers
relatively prime tom. ThenUmis a group under multiplication (modulom). Figure B-3 gives the multipli-
cation table forU 12 ={ 1 , 5 , 7 , 11 }.Fig. B-3 Fig. B-4Symmetric GroupSn
A one-to-one mappingσof the set {1, 2, ...,n} onto itself is called apermutation. Such a permutation may
be denoted as follows whereji=σ(i):
σ=(
123 ··· n
j 1 j 2 j 3 ··· jn)The set of all such permutations is denoted bySn, and there aren!=n(n− 1 )·...· 2 ·1 of them. The
composition and inverses of permutations inSnbelong toSn, and the identity functionεbelongs toSn. ThusSn
forms a group under composition of functions called thesymmetric group of degree n.
The symmetric groupS 3 has 3!=6 elements as follows:
ε=(
123
123)
,σ 2 =(
123
321)
,φ 1 =(
123
231)σ 1 =(
123
132)
,σ 3 =(
123
213)
,φ 2 =(
123
312)The multiplication table ofS 3 appears in Fig. B-4.