Download 3.1 Definition of a Group

3.1
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J.A.Beachy
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Definition of a Group
from A Study Guide for Beginner’s by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
This section contains the definitions of a binary operation, a group, an abelian group,
and a finite group. These definitions provide the language you will be working with, and
you simply must know this language. Try to learn it so well that you don’t have even a
trace of an accent!
Loosely, a group is a set on which it is possible to define an associative binary operation
that has an identity element, and which includes inverses for each of its elements. The
precise statement is given in Definition 3.1.4; 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 states that you can multiply both sides of an equation by equal quantities, 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 3.1.7 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 from high school algebra when working with 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.
Here are a few of the important examples. First, the sets of numbers Z, Q, R, and
C form groups under addition. Next, the sets Q× , R× , and C× of nonzero numbers form
groups under multiplication. The sets Z and Zn are groups under addition, while Z×
n
is a group under multiplication. It is common to just list these sets as groups, without
mentioning their operations, since in each case only one of the two familiar operations can
be used to make the set into a group. Similarly, the set Mn (R) of all n × n matrices with
entries in R is a group under addition, but not multiplication, while the set GLn (R) of
all invertible n × n matrices with entries in R is a group under multiplication, but not
under addition. There shouldn’t be any confusion in just listing these as groups, without
specifically mentioning which operation is used.
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.
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We give multiplication tables for three different groups of order 8. Various problems in
this section and later sections refer to these groups.
Table 3.1: Multiplication Table for O
e
a
a2
a3
b
ab
a2 b
a3 b
e
a
a2 a3
b ab a2 b a3 b
e
a
a2 a3
b ab a2 b a3 b
2
3
a
a
a
e ab a2 b a3 b
b
2
3
a
a
e
a a2 b a3 b
b ab
a3 e
a
a2 a3 b
b ab a2 b
2
3
b ab a b a b e
a
a2 a3
ab a2 b a3 b
b a
a2 a3 e
2
3
2
a b a b
b ab a
a3 e
a
a3 b
b ab a2 b a3 e
a
a2
Table 3.2: Multiplication Table for P
e
a
a2
a3
b
ab
a2 b
a3 b
e
a
a2 a3
b ab a2 b a3 b
e
a
a2 a3
b ab a2 b a3 b
2
3
a
a
a
e ab a2 b a3 b
b
2
3
a
a
e
a a2 b a3 b
b ab
a3 e
a
a2 a3 b
b ab a2 b
3
2
b a b a b ab e
a3 a2 a
ab
b a3 b a 2 b a
e
a3 a2
2
3
2
a b ab
b a b a
a
e
a3
3
2
3
2
a b a b ab
b a
a
a
e
SOLVED PROBLEMS: §3.1
25. Use the dot product to define a multiplication on R3 . Does this make R3 into a
group?
26. For vectors (x1 , y1 , z1 ) and (x2 , y2 , z2 ) in R3 , the cross product is defined by
(x1 , y1 , z1 )×(x2 , y2 , z2 ) = (y1 z2 − z1 y2 , z1 x2 − x1 z2 , x1 y2 − y1 x2 ) .
Is R3 a group under this multiplication?
27. On the set G = Q× of nonzero rational numbers, define a new multiplication by
ab
a ∗ b = , for all a, b ∈ G. Show that G is a group under this multiplication.
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Table 3.3: Multiplication Table for Q
e
a
a2
a3
b
ab
a2 b
a3 b
e
a
a2 a3
b ab a2 b a3 b
2
3
e
a
a
a
b ab a2 b a3 b
2
3
a
a
a
e ab a2 b a3 b
b
2
3
2
3
a
a
e
a a b a b
b ab
a3 e
a
a2 a3 b
b ab a2 b
b a3 b a2 b ab a2 a
e
a3
3
2
3
2
ab
b a b a b a
a
a
e
a2 b ab
b a3 b e
a3 a2 a
a3 b a2 b ab
b a
e
a3 a2
28. Write out the multiplication table for Z×
18 .
29. Write out the multiplication table for Z×
15 .
30. 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.
31. Let G be a group, and suppose that a and b are any elements of G. Show that
(aba−1 )n = abn a−1 , for any positive integer n.
32. In Definition 3.1.4, replace condition (iii) with the condition that there exists e ∈ G
such that e · a = a for all a ∈ G, and replace condition (iv) with the condition that
for each a ∈ G there exists a′ ∈ G with a′ · a = e. Prove that these weaker conditions
(given only on the left) still imply that G is a group.
33. Problem 32 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.
34. Let F be the set of all fractional linear transformations of the complex plane.
az + b
, where
That is, F is the set of all functions f (z) : C → C of the form f (z) =
cz + d
the coefficients a, b, c, d are integers with ad − bc = 1. Show that F forms a group if
we take the operation to be composition of functions.
35. Let G = {x ∈ R | x > 1} be the set of all real numbers greater than 1. For x, y ∈ G,
define x ∗ y = xy − x − y + 2.
(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 ∈ G there exists an inverse a−1 ∈ G.
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MORE PROBLEMS: §3.1
36.† For each binary operation ∗ given below, determine whether or not ∗ defines a group
structure on the given set. If not, list the group axioms that fail to hold.
(a) Define ∗ on Z by a ∗ b = min{a, b}.
(b) Define ∗ on Z+ by a ∗ b = min{a, b}.
(c) Define ∗ on Z+ by a ∗ b = max{a, b}.
37.† For each binary operation ∗ given below, determine whether or not ∗ defines a group
structure on the given set. If not, list the group axioms that fail to hold.
(a) Define ∗ on R by x ∗ y = x + y − 1.
(b) Define ∗ on R× by x ∗ y = xy + 1.
1 1
(c) Define ∗ on Q+ by x ∗ y = + .
x y
38. For each binary operation ∗ given below, determine whether or not ∗ defines a group
structure on the given set. If not, list the group axioms that fail to hold.
(a) Define ∗ on Z by x ∗ y = x2 y 3 .
(b) Define ∗ on Z+ by x ∗ y = 2xy .
(c) Define ∗ on Z+ by x ∗ y = xy .
39.† For each binary operation ∗ given below, determine whether or not ∗ defines a group
structure on the given set. If not, list the group axioms that fail to hold.
x y x, y ∈ R .
(a) Use matrix multiplication to define ∗ on
0 0 (b) Define ∗ on R2 by (x1 , y1 ) ∗ (x2 , y2 ) = (x1 x2 , y1 x2 + y2 ).
40. Let G be a group, and suppose that a, b, c ∈ G. Solve the equation axc = b.
41. In each of the groups O, P , Q in the tables in Section 3.1 of the Study Guide, find
the inverse of each of the eight elements.
42. In each of the groups O, P , Q in the tables in Section 3.1 of the Study Guide, solve
the equations ax = b and bx = a.
43. Let G be a group with operation · and let a ∈ G. Define a new operation ∗ on the set
G by x ∗ y = x · a · y, for all x, y ∈ G. Show that G is a group under the operation ∗.
Note: Problem 27 is a special case of this result.
44.† Let G be a group, with operation ·. Define a new operation on G by setting x ∗ y = y,
for x, y ∈ G. Determine which group axioms hold for this operation.
45. Let C[0, 1] be the set of all continuous functions from [0, 1] to R. Show that C[0, 1] is
a group under addition of functions: [f + g](x) = f (x) + g(x), for f (x), g(x) ∈ C[0, 1].
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46. Let G be a nonempty finite set with an associative binary operation ·. Exercise 3.1.20
shows that if the cancellation law holds on both sides, then G is a group. Give an
example in which the cancellation law holds on one side, but G is not a group.
47. Let T be the set of functions tc,d : R2 → R2 defined by tc,d (x1 , x2 ) = (x1 + c, x2 + d),
where c, d are any elements of R. Show that T is a group under composition of
functions.
48. Let G be a group. For x, y ∈ G, define x ∼ y if there exists some element a ∈ G such
that y = axa−1 . Show that ∼ defines an equivalence relation on G.
Note: This is a general case of Exercise 2.3.15 in the text, where the problem is stated
for the group G = Sn .