EXERCISES

In Exercises 4.1 to 4.4, which of the relations are equivalence relations? Describe the equivalence classes of those relations which are equivalence relations.

4.1. The relation on P × P defined by (a, b) (c, d) if and only if a + d =

b + c.

4.2.The relation T on the set of continuous functions from R to R, where fTg if and only if f (3) = g(3).

4.3.The inclusion relation on the power set P(X).

4.4.The relation C on a group G, where aCb if and only if ab = ba.

4.5.Find the left and right cosets of H = {(1), (12), (34), (12) Ž (34)} in S4.

4.6.Let H be the subgroup of A4 that fixes 1. Find the left and right cosets of H in A4. Is H normal? Describe the left cosets in terms of their effect on the element 1. Can you find a similar description for the right cosets?

In Exercises 4.7 to 4.12, verify that each of the functions is well defined. Determine which are group morphisms, and find the kernels and images of all the morphisms. The element of Zn containing x is denoted by [x]n.

4.7.f : Z12 → Z12, where f ([x]12) = [x + 1]12.

4.8.f : C12 → C12, where f (g) = g3.

4.9.f : Z → Z2 × Z4, where f (x) = ([x]2, [x]4).

4.10.f : Z8 → Z2, where, f ([x]8) = [x]2.

4.11.f : C2 × C3 → C3, where f (hr , ks ) = (12)r Ž (123)s .

4.12.f : Sn → Sn+1, where f (π) is the permutation on {1, 2, . . . , n + 1} defined by f (π)(i) = π(i) if i n, and f (π)(n + 1) = n + 1.

4.13.If H is a subgroup of an abelian group G, prove that the quotient group G/H is abelian.

4.14.If H is a subgroup of G, show that g−1Hg = {g−1hg|h H } is a subgroup for each g G.

4.15.Prove that the subgroup H is normal in G if and only if g−1Hg = H for all g G.

4.16.If H is the only subgroup of a given order in a group G, prove that H is normal in G.

4.17.Let H be any subgroup of a group G. Prove that there is a one-to-one correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

4.18.Is the cyclic subgroup {(1), (123), (132)} normal in S4?

4.19.Is the cyclic subgroup {(1), (123), (132)} normal in A4?

4.20.Is {(1), (1234), (13) Ž (24), (1432), (13), (24), (14) Ž (23), (12) Ž (34)} normal in S4?

4.21.Find all the group morphisms from C3 to C4.

4.22.Find all the group morphisms from Z to Z4.

4.23.Find all the group morphisms from C6 to C6.

4.24.Find all the group morphisms from Z to D4.

In Exercises 4.25 to 4.29, which of the pairs of groups are isomorphic? Give reasons.

4.25.C60 and C10 × C6.

4.26.(P{a, b, c}, ) and C2 × C2 × C2.

4.29.Z4 × Z2 and ({±1, ±i, ±(1 + i)/√2, ±(1 − i)/√2}, ·).

4.30.If G × H is cyclic, prove that G and H are cyclic.n 2 6 4n

4.31. If π is an r-cycle in Sn, prove that ρ−1 Ž π Ž ρ is also an r-cycle for each

ρ Sn.

4.32.Find four different subgroups of S4 that are isomorphic to S3.

4.33.Find all the isomorphism classes of groups of order 10.

4.34.Find all the ten subgroups of A4 and draw the poset diagram under inclusion. Which of the subgroups are normal?

H }.

4.36.Show that Q/Z is an infinite group but that every element has finite order.

4.37.If G is a subgroup of Sn and G contains an odd permutation, prove that G contains a normal subgroup of index 2.

commutators. Show that G is a normal subgroup of G. This is called the commutator subgroup. Also prove that G/G is abelian.

4.39. Let C be the group of nonzero complex numbers under multiplication and let W be the multiplicative group of complex numbers of unit modulus. Describe C /W .

4.40. Show that K = {(1), (12) Ž (34), (13) Ž (24), (14) Ž (23)} is a subgroup of S4

isomorphic to the Klein 4-group. Prove that K is normal and that S4/K =

S3.

4.41. If K is the group given in Exercise 4.40, prove that K is normal in A4 and

that A4/K = C3. This shows that A4 is not a simple group.

4.42.The cross-ratio of the four distinct real numbers x1, x2, x3, x4 in that order is the ratio λ = (x2 − x4)(x3 − x1)/(x2 − x1)(x3 − x4). Find the subgroup K of S4, of all those permutations of the four numbers that preserve the value of the cross-ratio. Show that if λ is the cross-ratio of four numbers taken in a certain order, the cross-ratio of these numbers in any other order must belong to the set

Furthermore, show that all permutations in the same coset of K in S4 give rise to the same cross-ratio. In other words, prove that the quotient group S4/K is isomorphic to the group of functions given in Exercise 3.42. The cross-ratio is very useful in projective geometry because it is preserved under projective transformations.

4.43.(Second Isomorphism Theorem) Let N be a normal subgroup of G, and let H be any subgroup of G. Show that H N = {hn|h H, n N } is a subgroup of G and that H ∩ N is a normal subgroup of H . Also prove that

4.44.(Third isomorphism theorem) Let M and N be normal subgroups of G, and N be a normal subgroup of M. Show that φ: G/N → G/M is a well-defined morphism if φ(Ng) = Mg, and prove that

(G/N )/(M/N ) = G/M.

4.45.If a finite group contains no nontrivial subgroups, prove that it is either trivial or cyclic of prime order.

4.46.If d is a divisor of the order of a finite cyclic group G, prove that G contains a subgroup of order d.

4.47. If G is a finite abelian group and p is a prime such that gp = e, for all g G, prove that G is isomorphic to Znp, for some integer n.

4.48.What is the symmetry group of a rectangular box with sides of length 2, 3, and 4 cm?

4.49.Let

If p is prime, show that (Gp, ·) is a group of order p(p2 − 1), and find a group isomorphic to G2.

4.50.Show that (R , ·) acts on Rn+1 by scalar multiplication. What are the orbits under this action? The set of orbits, excluding the origin, form the n-dimensional real projective space.

4.51.Let G be a group of order n, and let gcd(n, m) = 1. Show that every element h in G has an mth root; that is, h = gm for some g G.

4.52.Let G denote the commutator subgroup of a group G (see Exercise 4.38). If K is a subgroup of G, show that G H if and only if K is normal in G and G/K is abelian.

4.53.Call a group G metabelian if it has a normal subgroup K such that both K and G/K are abelian.

(a)Show that every subgroup and factor group of a metabelian group is metabelian. (Exercises 4.43 and 4.44 are useful.)

(b)Show that G is metabelian if and only if the commutator group G is abelian (see Exercise 4.38).

4.54.Recall (Exercise 3.76) that the center Z(G) of a group G is defined by Z(G) = {z G|zg = gz for all g G}. Let K Z(G) be a subgroup.

(a)Show that K is normal in G.

(b)If G/K is cyclic, show that G is abelian.

For Exercises 4.55 to 4.61, let Zm = {[x] Zm|gcd(x, m) = 1 }. The number of elements in this set is denoted by φ(m) and is called the Euler φ-function. For example, φ(4 ) = 2 , φ(6 ) = 2 , and φ(8 ) = 4 .

4.55.Show that φ(pr ) = pr − pr−1 if p is a prime.

4.56.Show that φ(mn) = φ(m)φ(n) if gcd(m, n) = 1.

4.57.Prove that (Zm , ·) is an abelian group.

4.58.Write out the multiplication table for (Z8 , ·).

4.59.Prove that (Z6 , ·) and (Z17 , ·) are cyclic and find generators.

4.60.Find groups in Table 4.6 that are isomorphic to (Z8 , ·), (Z9 , ·), (Z10 , ·), and (Z15 , ·) and describe the isomorphisms.

4.61.Prove that if gcd(a, m) = 1, then aφ(m) ≡ 1 mod m. [This result was known to Leonhard Euler (1707–1783).]

4.62.Prove that if p is a prime, then for any integer a, ap ≡ a modp. [This result was known to Pierre de Fermat (1601–1665).]

4.63.If G is a group of order 35 acting on a set with 13 elements, show that G

must have a fixed point, that is, a point x S such that g(x) = x for all g G.

4.64.If G is a group of order pr acting on a set with m elements, show that G has a fixed point if p does not divide m.