Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 3.1 Definition of a Group

 
Definition 3.1.1. A binary operation * on a set S is a function * : S × S -> S defined on the set S × S of all ordered pairs of elements in S and taking values in S.
      The operation * is said to be associative if

a * (b * c) = (a * b) * c

for all a,b,c in S.
      An element e in S is called an identity element for * if

a * e = a     and     e * a = a

for all a in S.
      If * has an identity element e, and a is an element of S, then an element b in S is said to be an inverse element for a if

a * b = e     and     b * a = e.

 
Proposition 3.1.2. Let * be an associative, binary operation on a set S.

(a) The operation * has at most one identity element.

(b) If * has an identity element, then any element of S has at most one inverse.

(c) If * has an identity element and elements a,b in S have inverses a-1 and b-1, respectively, then the inverse of a-1 exists and

( a-1 )-1 = a,

and the inverse of a * b exists and

( a * b )-1 = b-1 * a-1.

 
Definition 3.1.3. A group (G,·) is a nonempty set G together with a binary operation · on G such that the following conditions hold:

(i) Closure: For all a,b in G, the element a · b is a uniquely defined element of G.

(ii) Associativity: For all a,b,c in G, we have

a · (b · c) = (a · b) · c.

(iii) Identity: There exists an identity element e in G such that

e · a = a     and     a · e = a

for all a in G.

(iv) Inverses: For each a in G there exists an inverse element a-1 in G such that

a · a-1 = e     and     a-1 · a = e.

 
We will usually write ab for the product a · b.

 
Example 3.1.1. The set Q× of nonzero rational numbers, the set R× of nonzero real numbers, and the set C× of nonzero complex numbers form groups under ordinary multiplication.

 
Definition 3.1.4. The set of all permutations of a set S is denoted by Sym(S).
The set of all permutations of the set {1,2,...,n} is denoted by Sn.

 
Proposition 3.1.5. If S is any nonempty set, then Sym(S) is a group under the operation of composition of functions.

 
Proposition 3.1.6. (Cancellation Property for Groups) Let G be a group, and let a,b,c be elements of G.

(a) If ab = ac, then b = c.

(b) If ac = bc, then a = b.

 
Proposition 3.1.7. If G is a group and a,b belong to G, then the equations ax = b and xa = b have unique solutions.
      Conversely, if G is a nonempty set with an associative binary operation in which the equations ax = b and xa = b have solutions for all a,b in G, then G is a group.

 
Definition 3.1.8. A group G is said to be abelian if ab = ba for all elements a,b in G.

 
Definition 3.1.9. A group G is said to be a finite group if the set G has a finite number of elements. In this case, the number of elements is called the order of G, denoted by |G|.

 
Example 3.1.3. Zn is an abelian group under addition.

 
Example 3.1.4. Zn× is an abelian group under multiplication. Its order is given by the value (n) of Euler's phi-function.

 
Definition 3.1.10. The set of all invertible n × n matrices with entries in R is called the general linear group of degree n over the real numbers, and is denoted by GLn(R).

 
Proposition 3.1.11. The set GLn(R) forms a group under matrix multiplication.



§ 3.1 Definition of a Group: Solved problems

The definitions in this section provide the language you will be working with, and you simply must know this language. Loosely, a group is a set on which it is possible to define a binary operation that is associative, has an identity element, and has inverses for each of its elements. The precise statement is given in Definition 3.1.3; you must pay careful attention to each part, especially the quantifiers ("for all", "for each", "there exists"), which must be stated in exactly the right order.

From one point of view, the axioms for a group give us just what we need to work with equations involving the operation in the group. For example, one of the rules you are used to says that you can multiply both sides of an equation by the same value, and the equation will still hold. This still works for the operation in a group, since if x and y are elements of a group G, and x = y, then a ·: x = a · y, for any element a in G. This is a part of the guarantee that comes with the definition of a binary operation. It is important to note that on both sides of the equation, a is multiplied on the left. We could also guarantee that x · a = y · a, but we can't guarantee that a · x = y · a, since the operation in the group may not satisfy the commutative law.

The existence of inverses allows cancellation (see Proposition 3.1.6 for the precise statement). Remember that in a group there is no mention of division, so whenever you are tempted to write a ÷ b or a / b, you must write a · b-1 or b-1 · a. If you are careful about the side on which you multiply, and don't fall victim to the temptation to divide, you can be pretty safe in doing the familiar things to an equation that involves elements of a group.

Understanding and remembering the definitions will give you one level of understanding. The next level comes from knowing some good examples. The third level of understanding comes from using the definitions to prove various facts about groups.

In the study of finite groups, the most important examples come from groups of matrices. I should still mention that the original motivation for studying groups came from studying sets of permutations, and so the symmetric group Sn still has an important role to play.


22. Use the dot product to define a multiplication on R3. Does this make R3 into a group?     Solution

23. For vectors (a1,a2,a3) and (b1,b2,b3) in R3, the cross product is defined by

(a1,a2,a3) × (b1,b2,b3) = (a2b3-b3a2, a3b1-a1b3, a1b2-a2b1).

Is R3 a group under this multiplication?     Solution

24. On the set G = Q× of nonzero rational numbers, define a new multiplication by

a * b = ab/2,     for all a,b in G.

Show that G is a group under this multiplication.     Solution

25. Write out the multiplication table for Z9×.     Solution

26. Write out the multiplication table for Z15×.     Solution

27. Let G be a group, and suppose that a and b are any elements of G. Show that if (ab)2 = a2 b2, then ba = ab.     Solution

28. Let G be a group, and suppose that a and b are any elements of G. Show that (aba-1)n = a bn a-1, for any positive integer n.     Solution

29. In Definition 3.1.3 of the text, replace condition (iii) with the condition that there exists e in G such that e · a = a for all a in G, and replace condition (iv) with the condition that for each a in G there exists a' in G with a' · a = e. Prove that these weaker conditions (given only on the left) still imply that G is a group.     Solution

30. The previous exercise shows that in the definition of a group it is sufficient to require the existence of a left identity element and the existence of left inverses. Give an example to show that it is not sufficient to require the existence of a left identity element together with the existence of right inverses.     Solution

31. Let F be the set of all fractional linear transformations of the complex plane. That is, F is the set of all functions

f(z) : C -> C,   with   f(z) = (az+b)/(cz+d),

where the coefficients a,b,c,d are integers with ad-bc = 1. Show that F forms a group under composition of functions.     Solution

32. Let G = { x in R | x > 1 } be the set of all real numbers greater than 1. Define

x * y = xy - x - y + 2,   for x, y in G.

    (a) Show that the operation * is closed on G.
    (b) Show that the associative law holds for *.
    (c) Show that 2 is the identity element for the operation *.
    (d) Show that for element a in G there exists an inverse a-1 in G.
    Solution


§ 3.1 Lab questions

To answer these questions experimentally, you can use the Groups15 Java applet written by John Wavrik of UCSD.

Lab 1. Exercise 3.1.4 in the text asks you to write out the addition table for Z8. Groups15 also gives a table for Z8, using letters instead of congruence classes. Check that the two patterns are the same.

Lab 2. Exercise 3.1.5 in the text asks you to write out the multiplication table for Z7×. This is an abelian group of order 6, so you can compare your multiplication table with the one for Z6 produced by Groups15. Is the pattern the same? If not, can you rearrange your list of elements so that the multiplication table does have the same pattern?

Lab 3. In the list of groups given by Groups15, find the smallest one that is not abelian. (This is related to Exercise 3.1.14 in the text, which asks you to prove that a nonabelian group must have at least 5 different elements.) In this group, find an example of two elements x and y for which

(xy)2 x2y2.

(This is related to Exercise 3.1.15 in the text, which asks you to prove that a group G is abelian if and only if

(ab)n=anbn

for all elements a and b in G and all positive integers n.)

Lab 4. In the group of order 12 called A4 in Groups15, pair up the elements with their inverses. Is any element its own inverse? Find an example of two elements x,y for which

(xy) -1 x -1y -1.

(This is related to Exercise 3.1.19 in the text, which asks you to prove that a group G is abelian if and only if

(ab) -1 = a -1b -1

for all elements a and b in G.)

Lab 5. Exercise 3.1.21 in the text asks you to prove that if G is a finite group with an even number of elements, then there must exist an element a in G (different from the identity e) for which a2 = e.
As experimental evidence pointing to the truth of the exercise, show the following for each group listed by Groups15:
if the order of the group is an even number, then there is a nontrivial solution to the equation x2 = e.
Also show that if the order of the group is an odd number, then the only solution to the equation x2 = e is the trivial solution x = e.
Hint: In the groups listed in Groups15, the element A is the identity. To solve x2 = A, look for A on the diagonal of the group table.


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