Section 9.1 Algebraic Structures
713
28. Let 3S, # 4 be a monoid with identity i, and let x [ S. An element x L
−1
in S is a left inverse of x if x L
−1 # x = i.
An element x R
−1
in S is a right inverse of x if x # x R
−1
= i.
a. Prove that if every element in a monoid 3S, # 4 has both a left inverse and a right inverse, then 3S, # 4 is a
group.
b. Let S be the set of all functions f such that f: N S N. Then S under function composition is a monoid.
Define a function f [ S by f(x) = 2x, x [ N. Then define a function g [ S by
g (x) = e
x∙2 if x [ N, x even
1
if x [ N, x odd
Prove that g is a left inverse for f. Also prove that f has no right inverse.
29. Let 3S, # 4 be a semigroup having a left identity i L (see Exercise 27) and the property that for every x [ S,
x has a left inverse y such that y # x = i L . Prove that 3S, # 4 is a group. (Hint: y also has a left inverse in S.)
30. An element of a semigroup 3S, # 4 is idempotent if x # x = x. Prove that a group contains exactly one
idempotent element.
31. Prove that if 3S, # 4 is a semigroup in which the linear equations a # x = b and x # a = b are solvable for any
a, b [ S, then 3S, # 4 is a group. (Hint: Use Exercise 29.)
32. Prove that a finite semigroup that satisfies the left and right cancellation laws is a group. (Hint: Use
Exercise 29.)
33. Prove that a group 3G, # 4 is commutative if and only if (x # y)
2
= x
2 # y
2
for each x, y [ G.
34. Prove that a group 3G, # 4 in which x # x = i for each x [ G is commutative.
35. Let 3G, # 4 be a commutative group with identity i. For a fixed positive integer k, let B k = 5x 0 x [ G, x
k
= i6.
Prove that 3B k , # 4 is a subgroup of 3G, # 4.
36. Let 3G, # 4 be a commutative group with subgroups 3S, # 4 and 3T, # 4. Let ST = 5s # t 0 s [ S, t [ T6. Prove that
3ST, # 4 is a subgroup of 3G, # 4.
37. a. Let 3G, # 4 be a group and let 3S, # 4 and 3T, # 4 be subgroups of 3G, # 4. Prove that 3S d T, # 4 is a subgroup
of 3G, # 4.
b. Will 3S c T, # 4 be a subgroup of 3G, # 4? Prove or give a counterexample.
38. For any group 3G, # 4, the center of the group is A = 5x [ G 0 x # g = g # x for all g [ G}.
a. Prove that 3A, # 4 is a subgroup of 3G, # 4
b. Find the center of the group of symmetries of an equilateral triangle, 3D 3 , + 4 (see Exercise 9).
c. Prove that G is commutative if and only if G = A.
d. Let x and y be members of G with x # y
−1 [ A. Prove that x # y = y # x.
39. a. Let S A denote the group of permutations on a set A, and let a be a specific element of A. Prove that the
set H a of all permutations in S A that map a to a forms a subgroup of S A .
b. If A has n elements, what is 0 H a 0 ?
40. a. Let 3G, # 4 be a group and A # G, A ∙ [. Prove that 3A, # 4 is a subgroup of 3G, # 4 if for each
x, y [ A, x # y
−1 [ A. This subgroup test is sometimes more convenient to use than the theorem on
subgroups.
b. Use the test of part (a) to work Exercise 35.
41. a. Let 3G, # 4 be any group with identity i. For a fixed a [ G, a
0
denotes i and a
−n
means (a
n
)
−1
. Let
A = 5a
z
0 z [ Z). Prove that 3A, # 4 is a subgroup of G. (Hint: Use Exercise 40.)
b. The group 3G, # 4 is a cyclic group if for some a [ G, A = 5a
z
0 z [ Z6 is the entire group G. In this
case, a is a generator of 3G, # 4. For example, 1 is a generator of the group 3Z, + 4; remember that
the operation is addition. Thus, 1
0
= 0, 1
1
= 1, 1
2
= 1 + 1 = 2, 1
3
= 1 + 1 + 1 = 3… .; 1
−1
=
(1)
−1
= −1; 1
−2
= (1
2
)
−1
= −2; 1
−3
= (1
3
)
−1
= −3, … . Every integer can be written as an integral
713
28. Let 3S, # 4 be a monoid with identity i, and let x [ S. An element x L
−1
in S is a left inverse of x if x L
−1 # x = i.
An element x R
−1
in S is a right inverse of x if x # x R
−1
= i.
a. Prove that if every element in a monoid 3S, # 4 has both a left inverse and a right inverse, then 3S, # 4 is a
group.
b. Let S be the set of all functions f such that f: N S N. Then S under function composition is a monoid.
Define a function f [ S by f(x) = 2x, x [ N. Then define a function g [ S by
g (x) = e
x∙2 if x [ N, x even
1
if x [ N, x odd
Prove that g is a left inverse for f. Also prove that f has no right inverse.
29. Let 3S, # 4 be a semigroup having a left identity i L (see Exercise 27) and the property that for every x [ S,
x has a left inverse y such that y # x = i L . Prove that 3S, # 4 is a group. (Hint: y also has a left inverse in S.)
30. An element of a semigroup 3S, # 4 is idempotent if x # x = x. Prove that a group contains exactly one
idempotent element.
31. Prove that if 3S, # 4 is a semigroup in which the linear equations a # x = b and x # a = b are solvable for any
a, b [ S, then 3S, # 4 is a group. (Hint: Use Exercise 29.)
32. Prove that a finite semigroup that satisfies the left and right cancellation laws is a group. (Hint: Use
Exercise 29.)
33. Prove that a group 3G, # 4 is commutative if and only if (x # y)
2
= x
2 # y
2
for each x, y [ G.
34. Prove that a group 3G, # 4 in which x # x = i for each x [ G is commutative.
35. Let 3G, # 4 be a commutative group with identity i. For a fixed positive integer k, let B k = 5x 0 x [ G, x
k
= i6.
Prove that 3B k , # 4 is a subgroup of 3G, # 4.
36. Let 3G, # 4 be a commutative group with subgroups 3S, # 4 and 3T, # 4. Let ST = 5s # t 0 s [ S, t [ T6. Prove that
3ST, # 4 is a subgroup of 3G, # 4.
37. a. Let 3G, # 4 be a group and let 3S, # 4 and 3T, # 4 be subgroups of 3G, # 4. Prove that 3S d T, # 4 is a subgroup
of 3G, # 4.
b. Will 3S c T, # 4 be a subgroup of 3G, # 4? Prove or give a counterexample.
38. For any group 3G, # 4, the center of the group is A = 5x [ G 0 x # g = g # x for all g [ G}.
a. Prove that 3A, # 4 is a subgroup of 3G, # 4
b. Find the center of the group of symmetries of an equilateral triangle, 3D 3 , + 4 (see Exercise 9).
c. Prove that G is commutative if and only if G = A.
d. Let x and y be members of G with x # y
−1 [ A. Prove that x # y = y # x.
39. a. Let S A denote the group of permutations on a set A, and let a be a specific element of A. Prove that the
set H a of all permutations in S A that map a to a forms a subgroup of S A .
b. If A has n elements, what is 0 H a 0 ?
40. a. Let 3G, # 4 be a group and A # G, A ∙ [. Prove that 3A, # 4 is a subgroup of 3G, # 4 if for each
x, y [ A, x # y
−1 [ A. This subgroup test is sometimes more convenient to use than the theorem on
subgroups.
b. Use the test of part (a) to work Exercise 35.
41. a. Let 3G, # 4 be any group with identity i. For a fixed a [ G, a
0
denotes i and a
−n
means (a
n
)
−1
. Let
A = 5a
z
0 z [ Z). Prove that 3A, # 4 is a subgroup of G. (Hint: Use Exercise 40.)
b. The group 3G, # 4 is a cyclic group if for some a [ G, A = 5a
z
0 z [ Z6 is the entire group G. In this
case, a is a generator of 3G, # 4. For example, 1 is a generator of the group 3Z, + 4; remember that
the operation is addition. Thus, 1
0
= 0, 1
1
= 1, 1
2
= 1 + 1 = 2, 1
3
= 1 + 1 + 1 = 3… .; 1
−1
=
(1)
−1
= −1; 1
−2
= (1
2
)
−1
= −2; 1
−3
= (1
3
)
−1
= −3, … . Every integer can be written as an integral