Figure 3.9. Symmetries of an equilateral triangle.

TABLE 3.7. Group S3

Example 3.30. Show that D3 is isomorphic to S3 and write out the table for the latter group.

Solution. D3 is the group of symmetries of an equilateral triangle, and any symmetry induces a permutation of the vertices. This defines a function f : D3 → S3. If σ, τ D3, then f (σ Ž τ ) is the induced permutation on the vertices, which is the same as f (σ ) Ž f (τ ). Hence f is a morphism. Figure 3.9 illustrates the six elements of D3 and their corresponding permutations. We shade the underside of the triangle and mark the corner near vertex 1 to illustrate how the triangle moves. To visualize this, imagine a triangular jigsaw puzzle piece and consider all possible ways of fitting this piece into a triangular hole. Any proper rotation will leave the white side uppermost, whereas an improper rotation will leave the shaded side uppermost.

The six permutations are all distinct; thus f is a bijection and an isomorphism between D3 and S3. The group table for S3 is given in Table 3.7.

EVEN AND ODD PERMUTATIONS

We are going to show that every permutation can be given a parity, even or odd. The definition derives from an action of each permutation σ in Sn on a

68 3 GROUPS

called the discriminant, defined to be the product of all terms (xi − xj ), where i < j . More formally,

For example, if n = 3, then D = (x1 − x2)(x1 − x3)(x2 − x3). Given a permutation σ Sn, we have

We are going to determine an easy method for deciding which is the case. A 2- cycle is called a transposition, and surprisingly, much of the discussion centers around determining the parity of these transpositions. We are going to show that every transposition is odd.

Let D denote the discriminant in n variables x1, x2, . . . , xn, and define

Dk/m = the product of all terms in D involving xk , except (xk − xm) Dk,m = the product of all terms in D involving neither xk nor xm.

For example, if n = 5, we have

D2/4 = (x1 − x2)(x2 − x3)(x2 − x5)

D4/2 = (x1 − x4)(x3 − x4)(x4 − x5)

D2,4 = (x1 − x3)(x1 − x5)(x3 − x5).

Then D factors as follows:

D = (xk − xm)Dk/mDm/k Dk,m.

Now fix a transposition τ = (k m) in Sn, where k < m. Since τ interchanges k and m, we see that

τ Dk/m = uDm/k where u = 1 or u = −1.

Since τ 2 is the identity permutation, we have

Dk/m = τ 2Dk/m = τ (τ Dk/m) = τ (uDm/k ) = u(τ Dm/k ).

Because u2 = 1, it follows that

τ Dm/k = uDk/m.

Since τ Dk,m = Dk,m, applying τ to D gives

τD = τ (xk − xm) · τ Dk/m · τ Dm/k · τ Dk,m

=(xm − xk ) · uDm/k · uDk/m · Dk,m

=−D

because u2 = 1. Hence τ is odd, and we have proved:

Proposition 3.31. Every transposition is odd.

With this we can determine the parity of an arbitrary permutation σ in Sn. The idea is to factor σ as a product of transpositions, and we begin with the cycles. The proof of the next result is a straightforward verification.

Proposition 3.32. Every r-cycle is a product of r − 1 transpositions (not necessarily disjoint); in fact,

a1 a2 · · · ar = a1 a2 Ž a2 a3 Ž · · · Ž ar−1 ar .

Since every permutation σ is a product of disjoint cycles by Proposition 3.27, it follows that σ is a product of transpositions. This gives us the desired parity test.

Theorem 3.33. Parity Theorem. Every permutation σ Sn is a product of transpositions. Moreover, if σ is a product of m transpositions in any way at all, the parity of σ equals the parity of m. That is, σ is even if m is even, and σ is odd if m is odd.

The following result is now a consequence of Proposition 3.32.

Corollary 3.34. An n-cycle is an even permutation if n is odd and an odd permutation if n is even.

as a product of disjoint cycles and determine its order and parity.

Denote the set of even permutations on n elements by An. It follows from Theorem 3.33 that An is a subgroup of Sn, called the alternating group on n elements. For example,

Theorem 3.36. |An| = 12 n! for every n 2.

Proof. Let On denote the set of odd permutations in Sn, so that Sn = An On and An ∩ On = Ø. Hence n! = |Sn| = |An| + |On|, so it suffices to show that |An| = |On|. We do this by finding a bijection f : An → On. To this end, write

Proposition 3.37. Every even permutation can be written as a product of 3-cycles (not necessarily disjoint).

Proof. By Theorem 3.33, an even permutation can be written as a product of an even number of transpositions. We show that any product of two transpositions is a product of 3-cycles. If these two transpositions are identical, their product is the identity. If the two transpositions have one element in common, say (ab) and (bc), their product (ab) Ž (bc) = (abc), a 3-cycle. If the two transpositions have no elements in common, say (ab) and (cd), we can write their product as

(ab) Ž (cd) = (ab) Ž (bc) Ž (bc) Ž (cd) = (abc) Ž (bcd),

Theorem 3.33 and Proposition 3.37 show, respectively, that Sn is generated by the 2-cycles and An is generated by the 3-cycles.

CAYLEY’S REPRESENTATION THEOREM

At the beginning of the nineteenth century, groups appeared only in a very concrete form, such as symmetry groups or permutation groups. Arthur Cayley (1821–1895) was the first mathematician to deal with groups abstractly in terms of axioms, but he showed that any abstract group can be considered as a subgroup of a symmetric group. Hence, in some sense, if you know all about symmetry groups and permutation groups, you know all about group theory. This result is analogous to Stone’s representation theorem for boolean algebras, which proves that any abstract boolean algebra can be considered as an algebra of subsets of a set.

Theorem 3.38. Cayley’s Theorem. Every group (G, ·) is isomorphic to a subgroup of its symmetric group (S(G), Ž ).

Proof. For each element g G, define πg : G → G by πg (x) = g · x. We show that πg is a bijection. It is surjective because, for any y G, πg (g−1 · y) = g · (g−1 · y) = y. It is injective because πg (x) = πg (y) implies that g · x = g · y, and so, by Proposition 3.7, x = y. Hence πg S(G).

Let H = {πg S(G)|g G}. We show that (H, Ž ) is a subgroup of (S(G), Ž ) isomorphic to (G, ·). In fact, we show that the function ψ: G → H by ψ(g) = πg is a group isomorphism. This is clearly surjective. It is also injective because ψ(g) = ψ(h) implies that πg = πh, and πg (e) = πh(e) implies that g = h.

It remains to show that ψ preserves the group operation. If g, h G, πg·h(x) = (g · h)(x) = g · (h · x) = πg (h · x) = (πg Ž πh)(x) and πg·h = πg Ž

Corollary 3.39. If G is a finite group of order n, then G is isomorphic to a subgroup of Sn.

Proof. This follows because S(G) is isomorphic to Sn.

This is not of very much practical value, however, because Sn has order n!, which is much larger than the order of G, in general.

EXERCISES

Construct tables for the groups designated in Exercises 3.1 to 3.4.

For Exercises 3.5 to 3.16, state which are groups and which are abelian groups. Give reasons for your answers.

3.12.(Mn(R) − {0}, ·), where 0 is the n × n zero matrix.

3.13.({1, ζ, ζ 2, ζ 3, . . . , ζ n−1}, ·), where ζ is a complex nth root of 1.

3.14.({e, a}, ), where e e = e and e a = a e = a a = a.

3.15.(R , ), where R = R − {0} and x y is xy if x > 0, and x/y if x < 0.

3.16.(Z, ), where m n is m + n if m is even, and m − n if m is odd.

3.17.Prove that in any group (G, ·), (a1 · · · an)−1 = an−1 · · · a1−1.

3.18.If k is an integer (positive or negative), prove that (a−1ba)k = a−1bk a in any group (G, ·).

3.19.If G is a group in which a2 = e, the identity for all a G, show that G is abelian.

3.20.Prove that G is abelian if and only if (ab)2 = a2b2 for all a, b G.

3.21.If a is not the identity in a group and a4b = ba5, prove that ab = ba.

3.22.Prove that the order of the element g−1 is the same as the order of g.

3.23.Prove that the order of the element ab is the same as the order of ba.

3.24.Prove that every image of a cyclic group is cyclic.

3.25.The gaussian integers comprise the set of complex numbers Z[i] = {a + ib|a, b Z}. Is the group (Z[i], +) a cyclic group?

For Exercises 3.26 to 3.33, describe the symmetry groups of the figures.

For Exercises 3.34 and 3.35, describe the symmetry groups of the frieze patterns. These patterns are repeated indefinitely in both directions.

3.36.Prove that the relation of being a subgroup is a partial order on the set of subgroups of a group G.

3.37.Draw the poset diagram of the subgroups of C6.

3.38.Draw the poset diagram of the subgroups of S3.

3.39.If H and K are subgroups of a group G, prove that H ∩ K is also a subgroup of G. Is H K necessarily a subgroup of G?

3.40. If f : G → H and g: H → K are group morphisms, show that g Ž f : G →

K is also a group morphism.

3.41.Find all the group morphisms from (Z, +) to (Q, +).

3.42. Show that the set {f1, f2, f3, f4, f5, f6} of functions R − {0, 1} → R − {0, 1} under composition is isomorphic to S3, where

3.43. Is (Z, +) isomorphic to (Q , ·), where Q = Q − {0}? Give reasons. 3.44. Is (R, +) isomorphic to (R+, ·), where R+ = {x R|x > 0}? Give reasons. 3.45. Find the orders of all the elements in A4.

3.46. Is A4 = D6? Give reasons.

3.47.Draw the table and find the order of all the elements in the group

({±1, ±i, ±j, ±k}, ·), where i2 = j 2 = k2 = −1, ij = k = −j i, j k = i = −kj , and ki = j = −ik. This is called the quaternion group Q of order 8.

complex kth root of unity.

3.50.Construct the table for the group generated by g and h, where g and h satisfy the relations g3 = h2 = e and gh = hg2.

3.51.Prove that a group of even order always has at least one element of order 2.

3.52.Find a subgroup of S7 of order 10.

3.53.Find a subgroup of S5 of order 3.

3.54.Find a subgroup of A4 isomorphic to the Klein 4-group.

Multiply out the permutations in Exercises 3.55 to 3.58.

For Exercises 3.59 to 3.62, write the permutations as a product of disjoint cycles. Find the order of each permutation and state whether the permutation is even or odd.

Find the permutations for Exercises 3.63 to 3.66.

For each polynomial in Exercises 3.67 to 3.69, find the permutations of the subscripts that leave the value of the polynomial unchanged. These will form subgroups of S4 , called the symmetry groups of the polynomials.

3.69.(x1 − x2)2 + (x2 − x3)2 + (x3 − x4)2 + (x4 − x1)2.

3.70.Describe the group of proper rotations of the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1) in R3.

group morphism.

3.72.If g and h are elements in a group G, show that g and h−1gh have the same order.

3.73.What is the number of generators of the cyclic group Cn?

3.74.Express (123) Ž (456) as the power of a single cycle in S6. Can you generalize this result?

number of perfect shuffles that have to be performed on a deck of 52 cards before the cards are back in their original position? If there were 50 cards, what would be the least number?

3.76.The center of a group G is the set Z(G) = {x G|xg = gx for all g G}. Show that Z(G) is an abelian subgroup of G.

3.77.Find the center of D3.

3.78.Find the center of D4.

3.79.Prove that Sn is generated by the elements (12), (23), (34), . . . , (n − 1 n).

3.80.Prove that Sn is generated by the elements (123 · · · n) and (12).

3.81.Prove that An is generated by the set {(12r)|r = 3, 4, . . . , n}.

3.82.The well-known 15-puzzle consists of a shallow box filled with 16 small squares in a 4 × 4 array. The bottom right corner square is removed, and the other squares are labeled as in Figure 3.10. By sliding the squares around (without lifting them up), show that the set of possible permutations that can be obtained with the bottom right square blank is precisely A15. (There is no known easy proof that all elements in A15 must occur.)

Figure 3.10. The 15-puzzle.

3.83. Which of the positions of the 15-puzzle shown in Figure 3.10 can be achieved?

3.84.An automorphism of a group G is an isomorphism from G to itself. Prove that the set of all automorphisms of G forms a group under composition.

3.85.Find the automorphism group of the Klein 4-group.

3.86.Find the automorphism group of C3.

3.87.Find the automorphism group of C4.

3.88.Find the automorphism group of S3.

3.89.A word on {x, y} is a finite string of the symbols x, x−1, y, y−1, where x and x−1 cannot be adjacent and y and y−1 cannot be adjacent; for example, xxy−1x and x−1x−1yxy are words. Let F be the set of such words together with the empty word, which is denoted by 1. The operation of concatenation

places one word after another. Show that F is a group under concatenation, where any strings of the form xx−1, x−1x, yy−1, y−1y are deleted in a

concatenated word. F is called the free group on two generators. Is F abelian? What is the inverse of x−1x−1yxy?