48 3 GROUPS

GROUPS AND SYMMETRIES

A group (G, ·) is a set G together with a binary operation · satisfying the following axioms.

(i) G is closed under the operation ·; that is, a · b G for all a, b G. (ii) The operation · is associative; that is, (a · b) · c = a · (b · c) for all

a, b, c G.

(iii) There is an identity element e G such that e · a = a · e = a for all a G.

(iv)Each element a G has an inverse element a−1 G such that a−1 · a = a · a−1 = e.

The closure axiom is already implied by the definition of a binary operation; however, it is included because it is often overlooked otherwise.

If the operation is commutative, that is, if a · b = b · a for all a, b G, the group is called commutative or abelian, in honor of the mathematician Niels Abel.

Let G be the set of complex numbers {1, −1, i, −i} and let · be the standard multiplication of complex numbers. Then (G, ·) is an abelian group. The product of any two of these elements is an element of G; thus G is closed under the operation. Multiplication is associative and commutative in G because multiplication of complex numbers is always associative and commutative. The identity element is 1, and the inverse of each element a is the element 1/a. Hence 1−1 = 1, (−1)−1 = −1, i−1 = −i, and (−i)−1 = i. The multiplication of any two elements of G can be represented by Table 3.1.

The set of all rational numbers, Q, forms an abelian group (Q, +) under addition. The identity is 0, and the inverse of each element is its negative. Similarly, (Z, +), (R, +), and (C, +) are all abelian groups under addition.

If Q , R , and C denote the set of nonzero rational, real, and complex numbers, respectively, (Q , ·), (R , ·), and (C , ·) are all abelian groups under multiplication.

For any set X, (P(X), ) is an abelian group. The group axioms follow from Proposition 2.3; the empty set, Ø, is the identity, and each element is its own inverse.

TABLE 3.1. Group {1, −1, i, −i}

Every group must have at least one element, namely, its identity, e. A group with only this one element is called trivial. A trivial group takes the form ({e}, ·), where e · e = e.

Many important groups consist of functions. Given functions f : X → Y and g: Y → Z, their composite g Ž f : X → Z is defined by

(g Ž f )(x) = g(f (x)) for all x X.

Composition is associative; that is, if h: Z → W , then h Ž (g Ž f ) = (h Ž g) Ž f . Indeed,

(h Ž (g Ž f ))(x) = h(g(f (x))) = ((h Ž g) Ž f )(x)

for all x X, as is readily verified. In particular, if X is a set, then Ž is an associative binary operation on the set of all functions f : X → X. Moreover, this operation has an identity: The identity function 1X : X → X is defined by

equivalently if f (f (x)) = x and f (f (x)) = x for all x X. This inverse is unique when it exists. For if f is another inverse of f , then

f = f Ž 1X = f Ž (f Ž f ) = (f Ž f ) Ž f = 1X Ž f = f .

When it exists (see Theorem 3.3) the inverse of f is denoted f −1.

Example 3.1. A translation of the plane R2 in the direction of the vector (a, b) is a function f : R2 → R2 defined by f (x, y) = (x + a, y + b). The composition of this translation with a translation g in the direction of (c, d) is the function f Ž g: R2 → R2, where

f Ž g(x, y) = f (g(x, y)) = f (x + c, y + d) = (x + c + a, y + d + b).

composition. The identity is the identity transformation 1R2 : R2 → R2, and the inverse of the translation in the direction (a, b) is the translation in the opposite direction (−a, −b).

A function f : X → Y is called injective or one-to-one if f (x1) = f (x2) implies that x1 = x2. In other words, an injective function never takes two different points to the same point. The function f : X → Y is called surjective or

50 3 GROUPS

onto if for any y Y , there exists x X with y = f (x), that is, if the image f (X) is the whole set Y . A bijective function or one-to-one correspondence is a function that is both injective and surjective. A permutation or symmetry of a set X is a bijection from X to itself.

Lemma 3.2. If f : X → Y and g: Y → Z are two functions, then:

(i) If f and g are injective, g Ž f is injective.

(ii)If f and g are surjective, g Ž f is surjective.

(iii)If f and g are bijective, g Ž f is bijective.

Proof. (i) Suppose that (g Ž f )(x1) = (g Ž f )(x2). Then g(f (x1)) = g(f (x2)) so, since g is injective, f (x1) = f (x2). Since f is also injective, x1 = x2, proving that g Ž f is injective.

(ii) Let z Z. Since g is surjective, there exists y Y with g(y) = z, and since f is also surjective, there exists x X with f (x) = y. Hence (g Ž f )(x) =

The following theorem gives a necessary and sufficient condition for a function to have an inverse.

Theorem 3.3. Inversion Theorem. The function f : X → Y has an inverse if and only if f is bijective.

Proof. Suppose that h: Y → X is an inverse of f . The function f is injective because if f (x1) = f (x2), it follows that (h Ž f )(x1) = (h Ž f )(x2), and so x1 = x2. The function f is surjective because if y is any element of Y and x = h(y), it follows that f (x) = f (h(y)) = y. Therefore, f is bijective.

Conversely, suppose that f is bijective. We define the function h: Y → X as follows. For any y Y , there exists x X with y = f (x). Since f is injective, there is only one such element x. Define h(y) = x. This function h is an inverse to f because f (h(y)) = f (x) = y, and h(f (x)) = h(y) = x.

Theorem 3.4. If S(X) is the set of bijections from any set X to itself, then (S(X), Ž ) is a group under composition. This group is called the symmetric group or permutation group of X.

Proof. It follows from Lemma 3.1 that the composition of two bijections is a bijection; thus S(X) is closed under composition. The composition of functions is always associative, and the identity of S(X) is the identity function 1X : X → X. The inversion theorem (Theorem 3.3) proves that any bijective function f S(X) has an inverse f −1 S(X). Therefore, (S(X), Ž ) satisfies all the axioms for a group.

to use addition only for abelian groups. Furthermore, the identity under addition is usually denoted by 0 and the inverse of a by −a. Hence expressions of the form a · b−1 and an = a · · · a, in multiplicative notation, would be written as a − b and na = a + · · · + a, respectively, in additive notation.

In propositions and theorems concerning groups in general, it is conventional to use multiplicative notation and also to omit the dot in writing a product; therefore, a · b is just written as ab.

Whenever the operation in a group is clearly understood, we denote the group just by its underlying set. Therefore, the groups (Z, +), (Q, +), (R, +), and (C, +) are usually denoted just by Z, Q, R, and C, respectively. This should cause no confusion because Z, Q, R, and C are not groups under multiplication (since the element 0 has no multiplicative inverse). The symmetric group of X is denoted just by S(X), the operation of composition being understood. Moreover, if we refer to a group G without explicitly defining the group or the operation, it can be assumed that the operation in G is multiplication.

We now prove two propositions that will enable us to manipulate the elements of a group more easily. Recall from Proposition 2.10 that the identity of any binary operation is unique. We first show that the inverse of any element of a group is unique.

Proposition 3.6. Let be an associative binary operation on a set S that has identity e. Then, if an element a has an inverse, this inverse is unique.

Proof. Suppose that b and c are both inverses of a; thus a b = b a = e, and a c = c a = e. Now, since e is the identity and is associative,

b = b e = b (a c) = (b a) c = e c = c.

Hence the inverse of a is unique.

Note that if ab = e in a group G with identity e, then a−1 = b and b−1 = a. Indeed, b has an inverse b−1 in G, so b−1 = eb−1 = (ab)b−1 = ae = a. Similarly, a−1 = b.

Proposition 3.7. If a, b, and c are elements of a group G, then