(i) The cancellation laws hold: if a, b, x ∈ G, and either x ∗ a = x ∗ b or a∗ x = b ∗ x, then a = b.
(ii) (a−1)−1= a for all a ∈ G. (iii) If a, b∈ G, then
(a∗ b)−1= b−1∗ a−1. More generally, for all n≥ 2,
Proof. (i)
a= e ∗ a = (x−1∗ x) ∗ a = x−1∗ (x ∗ a)
= x−1∗ (x ∗ b) = (x−1∗ x) ∗ b = e ∗ b = b. A similar proof, using x∗ x−1 = e, works when x is on the right.
(ii) By Proposition 2.45(i), we have a∗ a−1 = e. But uniqueness of inverses, Proposition 2.45(iv), says that (a−1)−1is the unique x ∈ G such that x ∗a−1= e. Therefore, (a−1)−1= a.
(iii) By Lemma 2.43,
(a∗ b) ∗ (b−1∗ a−1)= [a ∗ (b ∗ b−1)] ∗ a−1= (a ∗ e) ∗ a−1= a ∗ a−1 = e. Hence, (a∗ b)−1 = b−1∗ a−1, by Proposition 2.45(iv). The second statement follows by induction on n≥ 2. •
In the proofs just given, we have been very careful about justifying every step and displaying all parentheses, for we are only beginning to learn the ideas of group theory. As one becomes more adept, however, the need for explicitly writing all such details lessens. This does not mean that one is allowed to become careless; it only means that one is growing. Of course, you must always be prepared to supply omitted details if your proof is challenged.
From now on, we will usually denote the product a∗ b in a group by ab (we have already abbreviated α◦ β to αβ in symmetric groups), and we will denote the identity by 1 instead of by e. When a group is abelian, however, we will often use additive notation. Here is the definition of group written in additive notation.
An additive group is a set G equipped with an operation+ and an identity element 0∈ G such that
(i) a+ (b + c) = (a + b) + c for every a, b, c ∈ G; (ii) 0+ a = a for all a ∈ G;
(iii) for every a ∈ G, there is −a ∈ G with (−a) + a = 0.
Note that the inverse of a, in additive notation, is written−a instead of a−1. We now give too many examples of groups (and there are more!). Glance over the list and choose several that look interesting to you.
(i) We remind the reader that SX, the set of all permutations of a set X , is a group under composition. In particular, Sn, the set of all permutations of
X= {1, 2, . . . , n}, is a group. (ii) The set
of all integers is an additive abelian group with a∗b = a+b, with identity e = 0, and with the inverse of an integer n being −n. Similarly, one can see that
, , and
are additive abelian groups. (iii) The set
×of all nonzero rationals is an abelian group, where∗ is ordinary
multiplication, the number 1 is the identity, and the inverse of r ∈
× is
1/r . Similarly,
× is a multiplicative abelian group. We show, in the next
example, that
× is also a multiplicative group.
Note that
×is not a group, for none of its elements (aside from±1)
has a multiplicative inverse in
×.
(iv) The nonzero complex numbers
× form an abelian group under multi-
plication. It is easy to see that multiplication is an associative operation and that 1 is the identity. Here is the simplest way to find inverses. If z= a + ib ∈
, where a, b∈
, define its complex conjugate z= a − ib. Note that zz= a2+ b2, so that z6= 0 if and only if zz 6= 0. If z 6= 0, then
z−1 = 1/z = z/zz = (a/zz) − (b/zz)i.
(v) The circle S1 of radius 1 with center the origin can be made into a mul- tiplicative abelian group if we regard its points as complex numbers of modulus 1. The circle group is defined by
S1= {z ∈
: |z| = 1},
where the operation is multiplication of complex numbers; that this is an operation on S1 follows from Corollary 1.20. Of course, complex multi- plication is associative, the identity is 1 (which has modulus 1), and the inverse of any complex number of modulus 1 is its complex conjugate, which also has modulus 1. Therefore, S1is a group. Even though S1is an abelian group, we still write it multiplicatively, for it would be confusing to write it additively.
(vi) For any positive integer n, let 0n =
ζk : 0 ≤ k < n be the set of all the nth roots of unity, where
ζ = e2π i /n= cos(2π/n) + i sin(2π/n).
The reader may use De Moivre’s theorem to see that 0nis an abelian group with operation multiplication of complex numbers; moreover, the inverse of any root of unity is its complex conjugate.
(vii) The plane ×
is an additive abelian group with operation vector addi- tion; that is, if v= (x, y) and v0= (x0,y0), then v+ v0= (x + x0,y+ y0). The identity is the origin O = (0, 0), and the inverse of v = (x, y) is −v = (−x, −y).
(viii) The parity group has two elements, the words “even” and “odd,” with operation
even + even = even = odd + odd and
even + odd = odd = odd + even. The reader may show that is an abelian group.
(ix) Let X be a set. Recall that if A and B are subsets of X , then their symmetric difference is A+ B = (A − B) ∪ (B − A) (symmetric difference is pictured in Figure 2.6). The Boolean group (X ) [named after the logician G. Boole (1815–1864)] is the family of all the subsets of X equipped with addition given by symmetric difference.
It is plain that A+ B = B + A, so that symmetric difference is com- mutative. The identity is
, the empty set, and the inverse of A is A itself, for A+ A =
. (See Exercise 2.3 on page 100.) Thus, (X ) is an abelian group.
Example 2.48.
(i) A (2× 2 real) matrix6 A is
A= a c b d , where a, b, c, d ∈ . If B= w y x z ,
then the product A B is defined by
A B= a c b d w y x z = aw+ cx ay+ cz bw+ dx by + dz .
6The word matrix (derived from the word meaning “mother”) means “womb” in Latin;
more generally, it means something that contains the essence of a thing. Its mathematical usage arises because a 2× 2 matrix, which is an array of four numbers, completely describes a certain type of function 2 → 2 called a linear transformation (more generally, larger matrices contain the essence of linear transformations between higher-dimensional spaces).
The elements a, b, c, d are called the entries of A. Call (a, c) the first row of A and call (b, d) the second row; call (a, b) the first column of A and call (c, d) the second column. Thus, each entry of the product A B is a dot product of a row of A with a column of B. The determinant of A, denoted by det( A), is the number ad− bc, and a matrix A is called nonsingular if det( A)6= 0. The reader may calculate that
det( A B)= det(A) det(B),
from which it follows that the product of nonsingular matrices is itself nonsingular. The set GL(2,
)of all nonsingular matrices, with operation matrix multiplication, is a (nonabelian) group, called the 2×2 real general linear group: the identity is the identity matrix
I =
1 0
0 1
and the inverse of a nonsingular matrix A is
A−1= d/1 −c/1 −b/1 a/1 ,
where 1 = ad − bc = det(A). (The proof of associativity is routine, though tedious; a “clean” proof of associativity can be given once one knows the relation between matrices and linear transformations [see Corol- lary 4.71].)
(ii) The previous example can be modified in two ways. First, we may allow the entries to lie in
or in
, giving the groups GL(2,
)or GL(2,
). We may even allow the entries to be in
, in which case GL(2,
)is defined to be the set of all such matrices with determinant±1 (one wants all the entries of A−1 to be in
). For readers familiar with linear algebra, all nonsingular n× n matrices form a group GL(n,
)under multiplication. (iii) All special7orthogonal matrices, that is, all matrices of the form
A= cos α − sin α sin α cos α ,
form a group denoted by S O(2,
), called the 2× 2 special orthogonal group. Let us show that matrix multiplication is an operation on S O(2,
). The product cos α − sin α sin α cos α cos β − sin β sin β cos β
is
cos α cos β− sin α sin β −[cos α sin β + sin α cos β] sin α cos β+ cos α sin β cos α cos β− sin α sin β
.
The addition theorem for sine and cosine shows that this product is again a special orthogonal matrix, for it is
cos(α+ β) − sin(α + β) sin(α+ β) cos(α+ β)
.
In fact, this calculation shows that S O(2,
) is abelian. It is clear that the identity matrix is special orthogonal, and we let the reader check that the inverse of a special orthogonal matrix (which exists because special orthogonal matrices have determinant 1) is also special orthogonal.
In Exercise 2.67 on page 166, we will see that S O(2,
) is a disguised version of the circle group S1, and that this group consists of all the rota- tions of the plane about the origin.
(iv) The affine8group Aff(1,
)consists of all functions →
(called affine maps) of the form
fa,b(x )= ax + b,
where a and b are fixed real numbers with a 6= 0. Let us check that Aff(1,
)is a group under composition. If fc,d(x )= cx + d, then fa,bfc,d(x )= fa,b(cx+ d)
= a(cx + d) + b = acx + (ad + b) = fac,ad+b(x ).
Since ac 6= 0, the composite is an affine map. The identity function 1 :
→
is an affine map (1 = f1,0), while the inverse of fa,b is easily seen to be fa−1,−a−1b. The reader should note that this composition
is reminiscent of matrix multiplication. a b 0 1 c d 0 1 = ac ad+ b 0 1 . Similarly, replacing by
gives the group Aff(1,
), and replacing by
gives the group Aff(1,
).
8Projective geometry involves enlarging the plane (and higher-dimensional spaces) by ad-
joining “points at infinity.” The enlarged plane is called the projective plane, and the original plane is called an affine plane. Affine functions are special functions between affine planes.
The following discussion is technical, and it can be skipped as long as the reader is aware of the statement of Theorem 2.49. Informally, this theorem says that if an operation is associative, then no parentheses are needed in products involving n≥ 3 factors.
An n-expression is an n-tuple (a1,a2, . . . ,an) ∈ G × · · · × G (n factors), and it yields many elements of G by the following procedure. Choose two adja- cent a’s, multiply them, and obtain an (n− 1)-expression: the new product just formed and n− 2 original a’s. In this shorter new expression, choose two adja- cent factors (either an original pair or an original one together with the new prod- uct from the first step) and multiply them. Repeat until a 2-expression (W, X ) is reached; now multiply and obtain the element W X in G. Call W X an ulti- mate product derived from the original expression. For example, consider the 4- expression (a, b, c, d). Let us multiply ab, obtaining the 3-expression (ab, c, d). We may now choose either adjacent pair ab, c or c, d; in either case, multiply these and obtain 2-expressions ((ab)c, d) or (ab, cd). The elements in either of these last expressions can now be multiplied to give the ultimate products [(ab)c]d or (ab)(cd). Other ultimate products derived from (a, b, c, d) arise by multiplying bc or cd as the first step, yielding (a, bc, d) or (a, b, cd). To say that an operation is associative is to say that the two ultimate products arising from 3-expressions (a, b, c) are equal. It is not obvious, even when an operation is associative, whether all the ultimate products derived from a longer expression are equal.
Definition. An n-expression (a1,a2, . . . ,an)needs no parentheses if all ulti- mate products derived from it are equal; that is, no matter what choices are made of adjacent factors to multiply, all the resulting products in G are equal.
Theorem 2.49 (Generalized Associativity). If n≥ 3, then every n-expression (a1,a2, . . . ,an)in a group G needs no parentheses.
Remark. Note that neither the identity element nor inverses will be used in the proof. Thus, the hypothesis of the theorem can be weakened by assuming that G is only a semigroup; that is, G is a nonempty set equipped with an associative binary operation.
Proof. The proof is by (the second form of) induction. The base step n = 3 follows from associativity. For the inductive step, consider 2-expressions of G obtained from an n-expression (a1,a2, . . . ,an)after two series of choices:
(W, X )= (a1· · · ai,ai+1· · · an) and (Y, Z )= (a1· · · aj,aj+1· · · an). We must prove that W X = Y Z in G. By induction, each of the elements W = a1· · · ai, X = ai+1· · · an, Y = a1· · · aj, and Z = aj+1· · · an, is the (one
and only!) ultimate product from m-expressions with m < n. Without loss of generality, we may assume that i ≤ j . If i = j , then the inductive hypothesis gives W = Y and X = Z in G, and so W X = Y Z, as desired.
We may now assume that i < j . Let A be the ultimate product from the i -expression (a1, . . . ,ai), let B be the ultimate product from the expression (ai+1, . . . ,aj), and let C be the ultimate product from the expression aj+1· · · an. The group elements A, B, and C are unambiguously defined, for the inductive hypothesis says that each of the shorter expressions yields only one ultimate product. Now W = A, for both are ultimate products from the i-expression (a1, . . . ,ai), Z = C [both are ultimate products from the (n − j )-expression (aj+1, . . . ,an)], X = BC [both are ultimate products from the (n−i)-expression (ai+1, . . . ,an)], and Y = AB [both are ultimate products from the j -expression (a1, . . . ,aj)]. We conclude that W X = A(BC) and Y Z = (AB)C, and so associativity, the base step n= 3, gives W X = Y Z, as desired. •
Definition. If G is a group and if a ∈ G, define the powers9 an, for n ≥ 1, inductively:
a1= a and an+1 = aan. Define a0= 1 and, if n is a positive integer, define
a−n = (a−1)n.
We let the reader prove that (a−1)n= (an)−1; this is a special case of the equa- tion in Lemma 2.46(iii).
There is a hidden complication here. The first and second powers are fine: a1 = a and a2 = aa. There are two possible cubes: we have defined a3 = aa2 = a(aa), but there is another reasonable contender: (aa)a = a2a. If one assumes associativity, then these are equal:
a3= aa2= a(aa) = (aa)a = a2a.
Generalized associativity shows that all powers of an elements are unambigu- ously defined.
9The terminology x square and x cube for x2and x3is, of course, geometric in origin.
Usage of the word power in this context goes back to Euclid, who wrote, “The power of a line is the square of the same line” (from the first English translation of Euclid, in 1570, by H. Billingsley). “Power” was the standard European rendition of the Greek dunamis (from which dynamo derives). However, contemporaries of Euclid, e.g., Aristotle and Plato, often used dunamis to mean amplification, and this seems to be a more appropriate translation, for Euclid was probably thinking of a 1-dimensional line sweeping out a 2-dimensional square. (I thank Donna Shalev for informing me of the classical usage of dunamis.)
Corollary 2.50. If G is a group, if a∈ G, and if m, n ≥ 1, then am+n= aman and (am)n= amn.
Proof. Both am+nand amanarise from the expression having m+n factors each equal to a; in the second instance, both (am)nand amn arise from the expression having mn factors each equal to a. •
It follows that any two powers of an element a in a group commute: aman= am+n = an+m = anam.
The proofs of the various statements in the next proposition, while straight- forward, are not short.
Proposition 2.51 (Laws of Exponents). Let G be a group, let a, b∈ G, and let m and n be (not necessarily positive) integers.
(i) If a and b commute, then (ab)n= anbn. (ii) (an)m= amn.
(iii) aman = am+n.
Proof. Exercises for the reader. •
The notation anis the natural way to denote a∗a∗· · ·∗a if a appears n times. However, if the operation is +, then it is more natural to denote a+ a + · · · + a by na. Let G be a group written additively; if a, b ∈ G and m and n are (not necessarily positive) integers, then Proposition 2.51 is usually rewritten:
(i) n(a+ b) = na + nb. (ii) m(na)= (mn)a. (iii) ma+ na = (m + n)a.
Example 2.52.
Suppose a deck of cards is shuffled, so that the order of the cards has changed from 1, 2, 3, 4, . . . , 52 to 2, 1, 4, 3, . . . , 52, 51. If we shuffle again in the same way, then the cards return to their original order. But a similar thing happens for any permutation α of the 52 cards: if one repeats α sufficiently often, the deck is eventually restored to its original order. One way to see this uses our knowledge of permutations. Write α as a product of disjoint cycles, say, α = β1β2· · · βt, where βiis an ri-cycle. Now βiri = (1) for every i, by Exercise 2.22 on page 120, and so βik = (1), where k = r1· · · rt. Since disjoint cycles commute, Exer- cise 2.27 on page 121 gives
Here is a more general result with a simpler proof (abstract algebra can be easier than algebra): if G is a finite group and a ∈ G, then ak = 1 for some k≥ 1. We use the argument in Lemma 2.23(i). Consider the sequence
1, a, a2, . . . ,an, . . . .
Since G is finite, there must be a repetition occurring in this sequence: there are integers m > n with am = an, and hence 1= ama−n = am−n. We have shown that there is some positive power of a equal to 1. Our original argument that αk = (1) for a permutation α of 52 cards is not worthless, for Proposition 2.54 will show that we may choose k to be the lcm(r1, . . . ,rt).
Definition. Let G be a group and let a∈ G. If ak = 1 for some k ≥ 1, then the smallest such exponent k ≥ 1 is called the order of a; if no such power exists, then one says that a has infinite order.
The argument given in Example 2.52 shows that every element in a finite group has finite order. In any group G, the identity has order 1, and it is the only element in G of order 1; an element has order 2 if and only if it is not the identity and it is equal to its own inverse. The matrix A= 1 10 1in the group GL(2,
) has infinite order, for Ak =1 k0 16=1 00 1for all k≥ 1.
Lemma 2.53. Let G be a group and assume that a ∈ G has finite order k. If an= 1, then k | n. In fact, {n ∈
: an = 1} is the set of all the multiples of k. Proof. It is easy to see that I = {n ∈
: an= 1} ⊆
satisfies the hypotheses of Corollary 1.34.
(i): 0∈ I because a0= 1.
(ii): If n, m ∈ I , then an = 1 and am = 1, so that an−m = ana−m = 1; hence,
n− m ∈ I .
(iii): If n∈ I and q ∈
, then an= 1 and aqn= (an)q = 1; hence, qn ∈ I . Therefore, I consists of all the multiples of k, where k is the smallest positive integer in I . But the smallest positive k in I is, by definition, the order of a. Therefore, if an= 1, then n ∈ I , and so n is a multiple of k. •
What is the order of a permutation in Sn?
Proposition 2.54. Let α∈ Sn.
(i) If α is an r -cycle, then α has order r .
(ii) If α = β1· · · βt is a product of disjoint ri-cycles βi, then α has order m = lcm(r1, . . . ,rt).
(iii) If p is a prime, then α has order p if and only if it is a p-cycle or a product of disjoint p-cycles.
Proof.
(i) This is Exercise 2.22(i) on page 120.
(ii) Each βi has order ri, by (i). Suppose that αM = (1). Since the βi commute, (1) = αM = (β1· · · βt)M = β1M· · · βtM. By Exercise 2.28(ii) on page 121, disjointness of the β’s implies that βiM = (1) for each i, so that Lemma 2.53 gives ri | M for all i; that is, M is a common multiple of r1, . . . ,rt. But if m = lcm(r1, . . . ,rt), then it is easy to see that αm = (1). Hence, α has order m. (iii) Write α as a product of disjoint cycles and use (ii). •
For example, a permutation in Sn has order 2 if and only if it is either a transposition or a product of disjoint transpositions.
We can now augment the table in Example 2.30.
Cycle Structure Number Order Parity
(1) 1 1 Even (1 2) 10 2 Odd (1 2 3) 20 3 Even (1 2 3 4) 30 4 Odd (1 2 3 4 5) 24 5 Even (1 2)(3 4 5) 20 6 Odd (1 2)(3 4) 15 2 Even 120 Table 2.3. Permutations in S5
Symmetry
We now present a connection between groups and symmetry. What do we mean when we say that an isosceles triangle 1 is symmetric? Figure 2.10 shows 1 = 1ABC with its base AB on the x-axis and with the y-axis being the perpendicular-bisector of A B. Close your eyes; let 1 be reflected in the y-axis (so that the vertices A and B are interchanged); open your eyes. You cannot tell that 1 has been reflected; that is, 1 is symmetric about the y-axis. On the other hand, if 1 were reflected in the x -axis, then it would be obvious, once your eyes are reopened, that a reflection had taken place; that is, 1 is not symmetric about the x -axis. Reflection is a special kind of isometry.
A
C
B
Figure 2.10 Isosceles Triangle
Definition. An isometry of the plane is a function ϕ: 2 →
2that is distance preserving: for all points P = (a, b) and Q = (c, d) in
2, kϕ(P) − ϕ(Q)k = kP − Qk,
wherekP − Qk =p(a− c)2+ (b − d)2is the distance from P to Q. Let P· Q denote the dot product:
P· Q = ac + bd. Now (P− Q) · (P − Q) = P · P − 2(P · Q) + Q · Q = (a2+ b2)− 2(ac + bd) + (c2+ d2) = (a2− 2ac + c2)+ (b2− 2bd + d2) = (a − c)2+ (b − d)2 = kP − Qk2.
It follows that every isometry ϕ preserves dot products: ϕ(P)· ϕ(Q) = P · Q, because
ϕ(P)· ϕ(Q) = kϕ(P) − ϕ(Q)k2= kP − Qk2 = P · Q. Recall the formula giving the geometric interpretation of the dot product:
where θ is the angle between P and Q. It follows that every isometry preserves angles. In particular, P and Q are orthogonal if and only if P· Q = 0, and so isometries preserve perpendicularity. Conversely, if ϕ preserves dot products, that is, if ϕ( P)· ϕ(Q) = P · Q, then the formula (P − Q) · (P − Q) = kP − Qk2 shows that ϕ is an isometry.
We denote the set of all isometries of the plane by Isom(
2); its subset consisting of all those isometries ϕ with ϕ(O) = O is called the orthogonal group of the plane, and it is denoted by O2(
). We will see, in Proposition 2.59, that both Isom(
2)and O 2(
)are groups under composition. We introduce some notation to help us analyze isometries.
Notation. If P and Q are distinct points in the plane, let L[P, Q] denote the line
they determine, and let PQ denote the line segment with endpoints P and Q. Here are some examples of isometries.
Example 2.55.
(i) Given an angle θ , rotation Rθ about the origin O is defined as follows:
Rθ(O) = O; if P 6= O, draw the line segment P O in Figure 2.11, rotate
it θ (counterclockwise if θ is positive, clockwise if θ is negative) to O P0, and define Rθ(P)= P0. Of course, one can rotate about any point in the
plane. P'= R (P) P O Figure 2.11 Rotation P L
.
(P) = P' L Figure 2.12 Reflection(ii) Reflection ρ`in a line `, called its axis, fixes each point in `; if P /∈ `, then ρ`(P)= P0, as in Figure 2.12 (` is the perpendicular-bisector of P P0). If
one pretends that the axis ` is a mirror, then P0is the mirror image of P. Now ρ`∈ Isom(
2); if ` passes through the origin, then ρ
`∈ O2(
(iii) Given a point V, translation10 by V is the function τV:
2 →
2 defined by τV(U ) = U + V . Translations lie in Isom(
2); a translation τV fixes the origin if and only if V = O, so that the identity is the only translation which is also a rotation.