MATH 335

Modern Algebra

Spring 2006

Homework

Week Homework
1/30-2/3 (1) Show that {1,2,3} under multiplication modulo 4 is not a group but that {1,2,3,4} under multiplication modulo 5 is a group.

(2) Show that the group GL(2,R) is non-Abelian by exhibiting a pair of matrices A and B in GL(2,R) such that AB is not equal BA.

(3) In a group, prove that (ab)-1 = b-1a-1. Find an example that shows that it is possible for (ab)-2 and b-2a-2 to be not equal. Find distinct nonidentity elements a and b from a non-Abelian group with the property that (ab)-1 = a-1 b-1. Draw an analogy between the statement (ab)-1 = b-1a-1 and the act of putting on and taking off your socks and shoes.

(4) Let a and b be elements of an Abelian group and let n be any integer. Show that (ab)n = anbn. Is this also true for non-Abelian groups?

Solutions

2/6-10 (1) Find elements A, B, and C in D4 such that AB = BC but A does not equal C.

(2) In Dn, let r denote clockwise rotation by 360/n degrees and let f be any reflection. Use a diagram to verify that frf = r-1. Use this relation to write the following elements in the form ri or rif, where 0 <= i < n.
(a) In D4, fr-2fr5.
(b) In D5, r-3fr4fr-2.
(c) In D6, fr5fr-2f.

(3) Prove that if G is a group with the property that the square of every element is the identity, then G is Abelian.

(4) For each group in the following list, find the order of the group and the order of each element in the group. What relation do you see between the orders of the elements of a group and the order of the group?
(a) Z12
(b) U(10)
(c) U(12)
(d) U(20)
(e) D4

Solutions

2/13-17 (1) Prove that in any group, an element and its inverse have the same order.

(2) Show that if a is an element of a group G, then |a| <= |G|.

(3) Let G be a group. Show that Z(G) equals the intersection of C(a), taken over all a in G.

(4) D4 has seven cyclic subgroups. List them. Find a subgroup of D4 of order 4 that is not cyclic.

Solutions

2/20-24 (1) How many subgroups does Z20 have? List a generator for each of these subgroups. Suppose that G = <a> and |a| = 20. How many subgroups does G have? List a generator for each of these subgroups.

(2) Suppose that |a| = 24. Find a generator for (<a21> intersected with <a10>). In general, what is a generator for the subgroup (<am> intersected with <an>)?

(3) Show that every group of order three is cyclic.

(4) Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z.

Solutions

2/27-3/3 (1) Show that a function from a finite set S to itself is one-to-one if and only if it is onto. Is this true when S is infinite?

(2) If a is even, prove that a-1 is even. If a is odd, prove that a-1 is odd.

(3) Let a =
[1 2 3 4 5 6 7 8]
[2 3 4 5 1 7 8 6]
and b =
[1 2 3 4 5 6 7 8]
[1 3 8 7 6 5 2 4].
Write a, b, and ab as
(a) products of disjoint cycles,
(b) products of 2-cycles.

(4) Compute the order of each member of A4. What arithmetic relationship do these orders have with the order of A4?

Solutions

3/6-10 (1) Let G be a group. For g in G, define the function Tg : G -> G by Tg(x) = gx. Prove that for any g in G, Tg is a permutation on the set of all elements of G. Show that Te is the identity map on G, and that for any g in G, Tg-1 = Tg-1.

(2) Show that U(8) is not isomorphic to U(10), and that U(8) is isomorphic to U(12).

(3) Prove that Z under addition is not isomorphic to Q under addition.

(4) Suppose that G is a finite Abelian group and G has no element of order 2. Show that the mapping g -> g2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism.

Solutions

3/13-17 (1) If G is a group, prove that Aut(G) and Inn(G) are groups.

(2) Suppose that g and h induce the same inner automorphism of a group G. Prove that h-1g is in Z(G).

(3) Find all the left cosets of {1,11} in U(30).

(4) Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime. (Do not assume at the outset that G is finite.)

Solutions

3/20-24 (1) Let G be a group of permutations on a set S, and fix x in S. Prove that StabG(x) is a subgroup of G.

(2) Let G be a group of permutations on a set S. Prove that the orbits of the members of S partition S.

(3) Show that GxH is Abelian if and only if G and H are Abelian.

(4) Find all subgroups of order 3 in Z9xZ3.

Solutions

3/27-29 (1) Let G = GL(2,R) and let K be a subgroup of R*. Prove that H = {A in G : det A in K} is a normal subgroup of G.

(2) Determine the order of (ZxZ)/<(2,2)>. Is this group cyclic?

(3) Let G = {1, -1, i, -i, j, -j, k, -k}, where i2 = j2 = k2 = -1, -i = (-1)i, 12 = (-1)2 = 1, ij = -ji = k, jk = -kj = i, and ki = -ik = j.
(a) Construct the Cayley table for G.
(b) Show that H = {1, -1} is a normal subgroup of G.
(c) Construct the Cayley table for G/H.

(4) Suppose that H is a normal subgroup of a finite group G. If G/H has an element of order n, show that G has an element of order n. Show, by example, that the assumption that G is finite is necessary. (Hint for the first part: show that, if aH has order n in G/H, then for any integer k, ak in H implies n|k).

Solutions

4/10-14 (1) Suppose that k is a divisor of n. Prove that Zn/<k> is isomorphic to Zk.

(2) How many homomorphisms are there from Z20 onto Z8? How many are there to Z8?

(3) Prove that the mapping phi: ZxZ -> Z given by phi(a,b) = a-b is a homomorphism. What is the kernel of phi? Describe the set phi-1(3).

(4) Prove the "Second" and "Third Isomorphism Theorems":
(a) If K is a subgroup of G and N is a normal subgroup of G, prove that K/(K intersect N) is isomorphic to KN/N.
(b) If M, N are normal subgroups of G and N is a normal subgroup of M, prove that (G/N)/(M/N) is isomorphic to G/M.

Solutions

4/17-21 (1) Find an integer n that shows that the rings Zn need not have the following properties that the ring of integers has.
(a) a2 = a implies a = 0 or a = 1.
(b) ab = 0 implies a = 0 or b = 0.
(c) ab = ac and a is not 0 imply b = c.
Show that the three properties are valid for Zp where p is prime.

(2) Show that a ring that is cyclic under addition is commutative.

(3) Let R be a ring. The center of R is the set {x in R : ax = xa for all a in R}. Prove that the center of a ring is a subring.

(4) Determine U(Z[i]), the units in the ring of Gaussian integers. (Hint: think about what the equation ac-bd = 1 says about gcd(a,b) and gcd(c,d).)

Solutions

4/24-28 (1) Show that every nonzero element of Zn is a unit or a zero-divisor.

(2) Let d be a positive integer. Prove that Z[sqrt(d)] is an integral domain, and that Q[sqrt(d)] is a field.

(3) If A and B are ideals of a ring, show that the product AB := {a1b1 + a2b2 + ... + anbn : aj in A, bj in B, n a positive integer} is an ideal. Find a positive integer a such that
(a) <a> = <3> <4>,
(b) <a> = <6> <8>,
(c) <a> = <m> <n>.

(4) Prove that the only ideals of a field F are {0} and F itself. Use this fact to show that R[x]/<x2+1> is a field.

Solutions

5/1-5 (1) Show that Z2[x]/< x2+x+1 > is a field. Make an addition and multiplication table for the field. Show that Z3[x]/< x2+x+1 > is not a field.

(2) Let phi be a ring homomorphism from R to S. Prove that ker(phi) = {r in R : phi(r) = 0} is an ideal of R.

(3) Let phi be a ring homomorphism from R to S. Prove that the mapping R/ker(phi) -> phi(R) given by r+ker(phi) -> phi(r) is an isomorphism.

(4) Prove that every ideal of a ring R is the kernel of a ring homomorphism by showing that an ideal I is the kernel of the mapping R -> R/I given by r -> r+I and that this mapping is a ring homomorphism.

Solutions

5/8-12 (1) Show that if m and n are distinct positive integers, then mZ is not ring-isomorphic to nZ.

(2) Determine the quotient and remainder upon dividing f(x) by g(x):
(a) f(x) = x3 + 2x + 4, g(x) = 3x + 2 in Z5[x];
(b) f(x) = 5x4 + 3x3 + 1, g(x) = 3x2 + 2x + 1 in Z7[x].

(3) Prove that the ideal <x> in Q[x] is maximal.

(4) Prove that Z[x] is not a principal ideal domain.

Solutions

"Data! Data! Data!", he cried, impatiently. "I can't make bricks without clay."
Sherlock Holmes (by Sir Arthur Conan Doyle, 1859-1930)

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