The New Chapter

Groups and Fields

Section G.1

Definitions and Examples of Groups

The following solutions make use of information found on pages 19-25 of the Reading Section.

Exercise G.1.1: Definitions

Fill in the blanks to complete the following sentences.

1. Recall from Chapter 7 that a binary operation is a special kind of function from X ×X to X.
2. A group (G, *) is a set G together with a binary operation * on G, such that:
   1. for every g,h G, g*h G.   (Closure)
   2. for every g,h,k G, (g*h)*k = g*(h*k).   (Associativity)
   3. there exists an element e G such that for every g G, g*e = g = e*g.   (Identity)
   4. for every g G, there exists g^-1 G such that g*g^-1 = e = g^-1*g.   (Inverse)
3. For any element g belonging to the group (G, *), the n^th power of g is given by gⁿ = g*g*g*... *g (n times) (where * is the binary operation of the group).
4. In a group (G, *) there exists one and only one identity element, e, such that for all g G, g*e = g = e*g.
   Similarly, for each g G there exists one and only one inverse element, g^-1, such that g*g^-1 = e = g^-1*g.
5. If the operation of the group (G, *) is commutative, then we say (G,*) is an abelian group. That is, it must satisfy all four of the group properties, as well as: for all g,h G, g*h = h*g.

Exercise G.1.2: Examples

You should attempt all these exercises yourself, using the workbook reading as an aid. Once you have attempted each question, check your answers by following the appropriate links. If you are stuck on a question, choose the link that gives you a hint and then try the question again.

1. Does the set {0,1} and the binary operation multiplication form a group? Explain.

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2. Show that (Z₇, +), where + denotes addition modulo 7, is a group. You may like to refer back to the Special Points in Chapter 10, Section 10.3 for the definition of Z₇.

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3. Verify that (Z₆-{0}, ×), where × denotes multiplication modulo 6, is not a group.

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4. If p is prime, then every non-zero element of Z_p has a multiplicative inverse in Z_p. Conversely, if every non-zero element of Z_p has a multiplicative inverse in Z_p, then p is prime.

Proof (part 1) Suppose p is prime and consider a Z_p, a 0. Since 0 < a < p, we know that gcd(a,p)=1, so we can use the Euclidean algorithm to find integers x and y such that ax + py = 1. Thus ax = 1 - py, so ax 1 (mod p). Thus [a]^-1 = [x].

Reverse the above argument to complete the rest of the proof.

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5. Find the multiplicative inverses of each of the non-zero elements of Z₇.

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6. Prove that (Z_p-{0}, ×), where p is prime and × denotes multiplication modulo p, is a group.

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7. Show that (Z,*), where * is defined by a*b = a+b+2,     for all a,b Z, is an abelian group.

Hint
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Section G.2

Elementary Properties of a Group

The following solutions make use of information found on pages 28-30 of the workbook Reading Section.

Exercise G.2.1: Definitions

Fill in the blanks to complete the following sentences.

1. Let G be a group under the binary operation *. If a subset H of G is itself a group under the group operation *, then H is said to be a subgroup of G. We denote this H G.
2. Let H be a subgroup of G. If H is not equal to the entire group G, then H is said to be a proper subgroup of G. If H is not equal to {e}, we say that H is a nontrivial subgroup of G. The subgroup {e} is said to be the trivial subgroup of G.
3. Let G be a group with binary operation * and identity element e. A subset H of G is a subgroup of G if and only if:
   1.e H   (identity);
   2. for all x H, x^-1 H   (inverse);
   3. for all x,y H, x * y H.   (closure).
4. ⁺ Let H be a subset of the group G. Then H is a subgroup of G if and only if H is nonempty and for all x,y H, xy^-1 H.
5. Let G be a group and a an element of G. Let < a > = {aⁿ G, for all  n Z}.
   The set < a > forms a subgroup and is called the cyclic subgroup of G generated by a. If < a > = G for some a G, then G is said to be cyclic and a is said to be a generator of G.
6. A group with a finite number of elements is said to be a finite group; otherwise it is an infinite group. A finite group G containing n elements is said to be of order n, written |G| = n or o(G) = n.
7. The order of an element g in G, written o(g), is defined to be the least positive integer n, if such an integer exists, such that gⁿ = e, where e is the identity element of G. If no such integer exists, then the order of the element is said to be infinite.
8. Let H be a subgroup of a group G and let g be an element of G. If H = {g^r | r Z}, then o(H) = o(g).

Exercise G.2.2: Examples

You should attempt all these exercises yourself, using the workbook Reading Section as an aid. Once you have attempted each question, check your answers by following the appropriate links. If you are stuck on a question, choose the link that gives you a hint and then try the question again.

1. In each of the following cases, determine whether H is a subgroup of G. Justify your answers.

a) H = {0,2,4,6,8};     G = (Z₁₀, +₁₀)     (where +₁₀ denotes addition modulo 10).

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b) H = (Z_n, +_n)     (where +_n denotes addition modulo n); G = (Z, +)     (where + denotes addition).

Hint
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c) H = {x R - {0} | x = 1 or x is irrational } with the binary operation ×; G = (R - {0}, ×)     (where × denotes multiplication in both cases).

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2.⁺ Let G be an abelian group with identity e. Use definition 4⁺ to show that H = {x G | x² = e} is a subgroup of G.

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3. Consider (Z₈, +₈) where +₈ denotes addition modulo 8. Find the cyclic subgroups generated by the elements [2] and [3]. What are o(2) and o(3)? Is either of these elements a generator of (Z₈, +₈)?

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4. A simple way of listing all the elements of a given finite group and their composition under the group operation is by using a Cayley table. A Cayley table for a group with n elements g₁,g₂, ..., g_n is an n ×n array with a headline and a sideline. The headline and sideline contain the elements of the group written in the same order. The entry in row i and column j of the body of the table is g_i * g_j for all i,j, where * is the binary operation of the group.

Consider the two groups given by the following Cayley tables. Notice that the second group is actually (Z₄, +₄) where +₄ denotes addition modulo 4.

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ii) What is the order of each element in each group?

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iii) Does either group contain any nontrivial subgroups? If so, what are they?

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iv) Is either of these groups cyclic? If so, name the generator(s).

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Section G.3

Definitions and Examples of Fields

The following solutions make use of information found on page 32 of the workbook Reading Section.

Exercise G.3.1:   Definitions

Fill in the blanks to complete the following sentences.

1. A field (F,+, *) is a set F together with two binary operations + and * on F, such that:
   1. (F, +) is an abelian group;
   2. (F - {0}, *) is an abelian group, where 0 denotes the additive identity;
   3. for a,b,c in F, a * (b + c) = a * b + a * c; that is, the distributive law holds in F.

Exercise G.3.2:   Examples

You should attempt all these exercises yourself, using the textbook as an aid. Once you have attempted each question, check your answers by following the appropriate links. If you are stuck on a question, choose the link that gives you a hint and then try the question again.

1. Verify that (Q, +, ·) is a field.

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2. Verify that (Z_p, +, ·), where p is a prime, is a field.

Hint
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3. Explain why (Z₆, +, ·) is not a field.

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