Math 345
These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence of a topic does not imply that it won't appear on the test.
1. Prove that if n is an integer, then is
either 1 or 2. Give specific examples which show that both cases can
occur.
2. Find the greatest common divisor of 847 and 133 and write it as a linear combination with integer coefficients of 847 and 133.
3. Show that the following set is a subgroup of :
However, show that it is not a normal subgroup of .
4. Consider the map given by
. Is
a group homomorphism? Why or why not?
5. Consider the map given by
. Is
a group homomorphism? Why or why
not?
6. is a group under componentwise
addition and
is a group under addition. Prove that
7. is a group under componentwise addition and
is a group under addition. Let
Prove that .
8. and
are groups under componentwise addition. Let
Show that
9. Here is the multiplication table for the Klein 4-group V:
Write down all the subgroups of V.
10. Find all integer solutions to
11. Find an element of order 30 in .
12. Find the primary decomposition of .
13. (a) What is the order of the element in the cyclic group
(b) What is the order of the element 10 in ?
(c) What elements generate the cyclic group ?
14. Subgroups of cyclic groups are cyclic. Give an example of an abelian group which is not cyclic, but in which every proper subgroup is cyclic.
15. (a) Prove that a group cannot be the union of two proper subgroups.
(b) Find a group which is a union of three proper subgroups.
16. Let be a group homomorphism. Prove that
is injective if and only if
.
17. Is there a group homomorphism such that
? Construct such a
homomorphism, or show that such a homomorphism cannot exist.
18. (a) Give an example of a finite group which is not abelian.
(b) Give an example of an abelian group which is not finite.
(c) Give an example of a group which is neither finite nor abelian.
19. Let denote the subgroup of
consisting of matrices of determinant 1. Show that
the following matrices lie in the same left coset of
:
20. Give an example of a finite commutative ring with 1 which is not an integral domain.
21. (a) Define by
Show that f is surjective.
(b) Define by
Show that g is surjective.
(c) Define by
Show that h is surjective.
(d) Define by
Show that k is not surjective.
(e) Give an example of a group map
which is not surjective, and a surjective function
which is not a group map.
22. (a) Explain why is not a group under
multiplication.
(b) Do the nonzero elements of form a group
under multiplication mod 6?
(c) Show that the nonzero elements of form a group
under multiplication mod 5. What group?
23. Reduce to an integer in the set
.
24. Reduce to an integer in
the set
. (Note: 149 is prime.)
25. The definition of a subring of a ring does not require that you check associativity for addition or multiplication. Explain why.
26. Prove that if I is an ideal in a ring R with identity and , then
.
27. Show that the only (two-sided) ideals in are the zero ideal and the whole ring.
28. Consider the following subset of the ring :
Check each axiom for an ideal. If the axiom holds, prove it. If the axiom does not hold, give a specific counterexample.
29. (a) Show that is irreducible in
.
(b) Find in
.
(c) Compute the product of the cosets in the quotient ring
. Write your answer
in the form
, where
.
30. Factor in
.
31. (a) Show that has no roots in
.
(b) Show that factors in
.
32. In the ring , consider the subset
(a) Show that is an ideal.
(b) Is in
?
33. is a factorization of
into irreducibles in
. Find a
different factorization of
into irreducibles in
.
34. Compute the product of the cycles
(right to left) and write the result as a product of disjoint cycles.
35. Define by
Determine which of the axioms for a ring map are satisfied by . If an axiom is not satisfied, give a specific
example which shows that the axiom is violated.
36. Define by
(a) Show that is a ring map.
(b) Determine the kernel of .
(c) Show that . Is
surjective?
37. Find the quotient and the remainder when is divided by
in
.
38. (a) Explain why has no roots in
.
(b) Is irreducible in
?
39. List the zero divisors and the units in .
40. Prove that if I is a left ideal in a division ring R, then either
or
.
41. Let R be a ring, and let . The
centralizer
of r is the set of elements of R which
commute with r:
Prove that is a subring of R.
42. Let
Prove that I is a left ideal, but not a right ideal, in the ring .
43. (a) List the elements of .
(b) List the elements of the subgroup in
.
(c) List the cosets of the subgroup in
.
(d) Is the quotient group isomorphic to or
?
44. Find the primary decomposition and the invariant factor
decomposition for .
45. What is the largest possible order of an element of ?
46. Let be group maps. Let
Prove that E is a subgroup of G. (E is called the equalizer of f and g.)
47. Let R be a ring such that for each , there is a
unique element
such that
. Prove that R has no zero divisors.
48. Suppose is a ring homomorphism and R and S
are rings with identity, but do not assume that
. Prove that if f is surjective, then
.
49. Factor in
.
50. Find the remainder when is divided by
in
.
51. Calvin Butterball thinks is
irreducible, based on the fact that solving
gives
, which are complex numbers.
Is he right?
52. Find the greatest common divisor of and
in
and express the greatest common divisor as
a linear combination (with coefficients in
) of the two polynomials.
53. The following set is an ideal in the ring :
(a) List the cosets of I in .
(b) Construct addition and multiplication tables for the quotient
ring .
(c) Is an integral
domain?
1. Prove that if n is an integer, then is
either 1 or 2. Give specific examples which show that both cases can
occur.
Note that
Now divides
and
, so it divides
, and hence it divides 2. The only positive
integers that divide 2 are 1 and 2. Hence,
is either 1 or 2.
If , I have
and
, and
.
If , I have
and
, and
.
This shows that both cases can occur.
2. Find the greatest common divisor of 847 and 133 and write it as a linear combination with integer coefficients of 847 and 133.
The greatest common divisor is 7, and
3. Show that the following set is a subgroup of :
However, show that it is not a normal subgroup of .
Since , H contains the identity.
If
, then
(Note that implies
and
, so
and
are defined.)
Therefore, H is closed under taking inverses.
Finally,
(If and
, then
, so
and
.) Thus, H is closed under products. Hence, H is a
subgroup.
However,
Therefore, H is not a normal subgroup.
4. Consider the map given by
. Is
a group homomorphism? Why or why not?
A group homomorphism must map the identity in the domain to the
identity in the range. The identity in is 0. However,
. Therefore,
is not a
homomorphism.
5. Consider the map
given by
. Is
a group homomorphism? Why or why not?
In this case, , so
does map the identity to the identity.
However,
Since for all a and b,
is not a homomorphism.
6. is a group under componentwise
addition and
is a group under addition. Prove that
Define by
f can be represented by matrix multiplication:
Hence, it's a group map.
Let . Then
Thus, .
Let . Then
Now but
. By Euclid's
lemma,
. Say
. Then
Therefore,
Thus, .
Hence, .
Let . Note that
Multiplying by z, I get
Then
This proves that .
Hence,
7. is a group under componentwise addition and
is a group under addition. Let
Prove that .
Define by
Note that
Since f can be expressed as multiplication by a constant matrix, it's a linear transformation, and hence a group map.
Let . Then
Therefore, , and
hence
.
Let . Then
Hence,
Therefore, . Hence,
.
Let . Note that
Hence, .
Thus,
8. and
are groups under componentwise addition. Let
Show that
Define by
Note that
Since f can be written as multiplication by a constant matrix, it is a group map.
Let . Then
Hence, , so
.
Let . Then
This gives and
. The first
equation gives
and the second equation gives
. Hence,
Therefore, , and hence
.
Let . Then
Hence, f is surjective, and .
Therefore,
9. Here is the multiplication table for the Klein 4-group V:
Write down all the subgroups of V.
By Lagrange's theorem, the order of a subgroup must divide the order of the group. Hence, there could be subgroups of order 1, 2, or 4.
The subgroup of order 1 is ; the subgroup of
order 4 is the whole group. A subgroup of order 2 must contain the
identity and another element; by closure under inverses, the other
element must be its own inverse. Hence, the subgroups of V are:
10. Find all integer solutions to
This equation expresses 11 as a product of two integers and
. There are four ways to do this.
Case 1.
Solving simultaneously, I get and
.
Case 2.
Solving simultaneously, I get and
.
Case 3.
Solving simultaneously, I get and
.
Case 4.
Solving simultaneously, I get and
.
The solutions are ,
,
, and
.
11. Find an element of order 30 in .
5 has order 5 in .
2 has order 6 in .
Hence, has order
in
.
12. Find the primary decomposition of .
The operation is multiplication mod 16. The possibilities are
I start computing the orders of elements. The order of an element can be 1, 2, 4, 8, or 16, so I can repeatedly square until I get the identity.
Since 3 has order 4, and since every element of has order 2 or less,
is ruled out.
Since there are no elements of order 8, the group can't be . Hence,
.
13. (a) What is the order of the element in the cyclic group
(b) What is the order of the element 10 in ?
(c) What elements generate the cyclic group ?
(a) The order of in the cyclic group of order n with
generator a is
. So the order of
in
is
(b) The order of 10 in is
(c) The order of the element is
. If m generates
, it must have order 12, so
This implies that ; that is, m is relatively prime
to 12. Therefore, the generators are
.
14. Subgroups of cyclic groups are cyclic. Give an example of an abelian group which is not cyclic, but in which every proper subgroup is cyclic.
V is not cyclic, since there are no elements of order 4. However,
every subgroup of V is cyclic.
15. (a) Prove that a group cannot be the union of two proper subgroups.
(b) Find a group which is a union of three proper subgroups.
(a) Suppose G is a group, H and K are proper subgroups, and . Since H is not all of G, I can find an element
such that
. Likewise, I can find
an element
such that
.
Now consider the element . It's in G, so it's either in H
or K. But
gives
, contradicting the assumption that
. And
gives
, which contradicts the assumption that
.
Therefore, G cannot be the union of H and K.
(b) Consider the Klein 4-group V:
V is the union of the proper subgroups ,
, and
.
16. Let be a group homomorphism. Prove that
is injective if and only if
.
Suppose that is injective. (This means that different
inputs go to different outputs, or alternatively, that
implies
.) I want to show
that
.
Since , I need to show
implies
. Therefore, take
, so
. Now
, so
. Since
is injective, this implies that
, which is what I wanted to show.
Conversely, suppose . I want to show that
is injective. To do this, suppose
. I need to show
. Rearrange the equation:
But this means that , i.e.
Therefore, is injective.
17. Is there a group homomorphism such that
? Construct such a
homomorphism, or show that such a homomorphism cannot exist.
If is a homomorphism
such that
, then
is 1-1. Since the image of
will be isomorphic to
, the image of such a map must be a cyclic subgroup
of order 6.
The only subgroup of order 6 in is
The only possibility is that maps
isomorphically onto this subgroup. Such an
isomorphism must send the generator
to a
generator of
. Since 2 generates
, I will try
.
Since is supposed to be a group map, this forces
for
. Then if
,
Hence, is a group map.
Finally, the only element of that maps to 0
is 0, by inspection. Thus,
, and
satisfies the conditions of the problem.
18. (a) Give an example of a finite group which is not abelian.
(b) Give an example of an abelian group which is not finite.
(c) Give an example of a group which is neither finite nor abelian.
(a) is finite, but not abelian.
(b) is abelian, but not finite.
(c) is an infinite group which is not abelian.
For example,
19. Let denote the subgroup of
consisting of matrices of determinant 1. Show that
the following matrices lie in the same left coset of
:
If H is a subgroup of a group G, then if and only if
. In this case,
Now
Hence,
This shows that the matrices lie in the same left coset of .
20. Give an example of a finite commutative ring with 1 which is not an integral domain.
is finite, commutative, and has a multiplicative
identity 1. But
, so it's not a domain.
21. (a) Define by
Show that f is surjective.
(b) Define by
Show that g is surjective.
(c) Define by
Show that h is surjective.
(d) Define by
Show that k is not surjective.
(e) Give an example of a group map
which is not surjective, and a surjective function
which is not a group map.
(a) Let . Then
Therefore, f is surjective.
(b) Let . Then
Therefore, g is surjective.
(c) Let . Then
Therefore, h is surjective.
(d) . But if
This contradiction shows that there is no such that
. Hence, k is
not surjective.
(e) The function defined by
is a group map, since
However, p is not surjective, since (for example) there is no such that
.
The function given by
is surjective: If
, then
But q is not a group map: , so q does not map the
identity to the identity.
For that matter, the identity map is a surjective group map, and the function
given by
is neither
surjective nor a group map. The properties of surjectivity and being
a group map are independent.
22. (a) Explain why is not a group under
multiplication.
(b) Do the nonzero elements of form a group
under multiplication mod 6?
(c) Show that the nonzero elements of form a group
under multiplication mod 5. What group?
(a) is not a group under multiplication because
not every element has a multiplicative inverse. To be specific,
does not have a multiplicative inverse.
(b) is not a group under multiplication mod 6,
because it is not closed under the operation:
, for instance.
(c) Here is the operation table:
The table shows that set is closed under the operation. Take for
granted that multiplication mod 6 is associative (since is a ring under addition and multiplication mod 6). 1
is the identity element. The inverse of 2 is 3, the inverse of 3 is
2, and 4 is its own inverse. Therefore, this set is a group; it's
usually denoted
.
a group with 4 elements. and the table
shows that not every element has order 2. Therefore,
is not isomorphic to
; it must be isomorphic to
.
23. Reduce to an integer in the set
.
, so by Fermat's theorem,
. Therefore,
Since ,
. Hence,
24. Reduce to an integer in
the set
. (Note: 149 is prime.)
At this point, you could use the Extended Euclidean Algorithm to find the inverses of 3 and 75 mod 149. But it's easier to note that
Since and
, I have
25. The definition of a subring of a ring does not require that you check associativity for addition or multiplication. Explain why.
When you consider a subset S of a ring R, addition and multiplication are associative as operations in R. In showing that S is a subring, you're confining the operations to a subset, so they must continue to be associative.
(People often say that associativity is
inherited from R by S.) For similar reasons, the definition of a
subgroup does not require that you check
associativity.
26. Prove that if I is an ideal in a ring R with identity and , then
.
Since by definition, I only need to prove the
opposite containment. Let
. Now
, so
, i.e.
. Hence,
, so
.
27. Show that the only (two-sided) ideals in are the zero ideal and the whole ring.
Let S be an ideal in , and suppose S is
nonzero. I'll show that
.
S contains a nonzero matrix A. If A is invertible, then implies
, i.e.
, where I is the identity matrix. By the last
problem, this implies that
.
Suppose then that A is not invertible. Any matrix row reduces to one of the following:
A is not invertible, so it doesn't row reduce to I; it's nonzero, so it doesn't row reduce to the zero matrix.
Suppose A row reduces to . There are elementary matrices
, ...,
such that
Since , this equation shows that
.
Since S is an ideal,
Again, since S is an ideal,
And again, since S is an ideal,
Hence,
Hence, .
A similar argument shows that if A row reduces to , then
.
Therefore, the only ideals in are the zero
ideal and the whole ring.
28. Consider the following subset of the ring :
Check each axiom for an ideal. If the axiom holds, prove it. If the axiom does not hold, give a specific counterexample.
The zero element is in S, since .
Let . Then
Let . Then
I have . Then
But . For suppose
for
. Then
Adding the two equations gives , but this equation has no
integer solutions.
Thus, S is not an ideal in
.
29. (a) Show that is irreducible in
.
(b) Find in
.
(c) Compute the product of the cosets in the quotient ring
. Write your answer
in the form
, where
.
(a) Since it's a quadratic, it suffices to show that it has no roots
in .
It has no roots in , so it's irreducible over
.
(b) In general, you can find an inverse using the Extended Euclidean
Algorithm. In this case, the coset representative is linear, so I can just Apply the Division
Algorithm:
Hence,
(c) First,
Apply the Division Algorithm:
Therefore, the product is
30. Factor in
.
The idea is to add a middle term to complete the square, then subtract it back off:
You can check using the Quadratic Formula that and
do not factor over
.
31. (a) Show that has no roots in
.
(b) Show that factors in
.
(a)
(b) .
32. In the ring , consider the subset
(a) Show that is an ideal.
(b) Is in
?
(a) Suppose , where
. Then
I have
Let . Then
Finally, let and let
. Then
(Note that is commutative, so I only need to check
multiplication on the left.) Hence,
is an ideal.
(b) The greatest common divisor of and
is
, and it must divide any linear combination
. Suppose then that
Then . But in fact,
Thus, , and so
cannot be an element of
.
33. is a factorization of
into irreducibles in
. Find a
different factorization of
into irreducibles in
.
Since in
,
34. Compute the product of the cycles
(right to left) and write the result as a product of disjoint cycles.
The product is .
35. Define by
Determine which of the axioms for a ring map are satisfied by . If an axiom is not satisfied, give a specific
example which shows that the axiom is violated.
First, .
If ,
However,
Thus, .
36. Define by
(a) Show that is a ring map.
(b) Determine the kernel of .
(c) Show that . Is
surjective?
(a) If ,
Therefore, is a ring map.
(b) means
, which is only
possible if
(since
is an
integral domain). Therefore,
.
(c)
is not surjective. If
, then
. This implies
, or
. Obviously,
. This
contradiction shows that x is not in the image of
, and
is not surjective.
37. Find the quotient and the remainder when is divided by
in
.
The quotient is and the remainder is 4.
38. (a) Explain why has no roots in
.
(b) Is irreducible in
?
(a) Since for all x, it follows that
. In particular, no real value of x makes
it 0.
(b)
Hence, is not irreducible in
.
39. List the zero divisors and the units in .
The zero divisors are ,
, and
.
The units are and
.
40. Prove that if I is a left ideal in a division ring R, then either
or
.
Suppose . Then I can find a nonzero element
. Since R is a division ring, x is invertible. Since
I is a left ideal,
. But
, so
. An ideal that contains 1
is the whole ring, so
.
41. Let R be a ring, and let . The
centralizer
of r is the set of elements of R which
commute with r:
Prove that is a subring of R.
Let , so
and
. Then
Therefore, .
Since , I have
.
Let , so
. Then
Hence, .
Let , so
and
. Then
Therefore, .
Hence, is a subring.
42. Let
Prove that I is a left ideal, but not a right ideal, in the ring .
Hence, I is closed under sums.
Elements of I are exactly the matrices with all-zero
first and third columns. Thus,
If
Thus, I is closed under taking additive inverses.
Let
Then
Hence, I is a left ideal.
However,
But
Hence, I is not a right ideal.
43. (a) List the elements of .
(b) List the elements of the subgroup in
.
(c) List the cosets of the subgroup in
.
(d) Is the quotient group isomorphic to or
?
(a)
(b)
(c)
(d) Note that
The results are all elements of the identity coset .
So all three cosets have order 2.
is isomorphic to
, since every element squares to
the identity.
44. Find the primary decomposition and the invariant factor
decomposition for .
Therefore, the primary decomposition is
Here's the work for the invariant factor decomposition:
The invariant factor decomposition is .
45. What is the largest possible order of an element of ?
The primary decomposition is
Compute the invariant factor decomposition:
The invariant factor decomposition is . Hence, the largest possible
order of an element is 180.
46. Let be group maps. Let
Prove that E is a subgroup of G. (E is called the equalizer of f and g.)
Let , so
and
. Then
Therefore, .
Since ,
.
Let , so
. Then
, so
.
Hence,
.
Therefore, E is a subgroup of G.
47. Let R be a ring such that for each , there is a
unique element
such that
. Prove that R has no zero divisors.
Suppose that is a zero divisor, so
and
for some
. Let s be the unique element of R such that
. Then
But was the unique solution to
, so
, and
. This contradiction implies that there is no such t,
so R has no zero divisors.
48. Suppose is a ring homomorphism and R and S
are rings with identity, but do not assume that
. Prove that if f is surjective, then
.
Since f is surjective, there is an element such that
. Then
49. Factor in
.
If a cubic or quadratic polynomial over a field factors, it must have
a linear factor, i.e. a root. Therefore, I'll try the elements of
to find the roots.
1, 2, and 3 are roots, so ,
, and
are factors. Since the
leading coefficient is 3, I must have
50. Find the remainder when is divided by
in
.
Notice that in
. By the
Remainder Theorem, the remainder is
51. Calvin Butterball thinks is
irreducible, based on the fact that solving
gives
, which are complex numbers.
Is he right?
In fact, since in
,
Thus, is not irreducible in
.
52. Find the greatest common divisor of and
in
and express the greatest common divisor as
a linear combination (with coefficients in
) of the two polynomials.
The greatest common divisor is , and
53. The following set is an ideal in the ring :
(a) List the cosets of I in .
(b) Construct addition and multiplication tables for the quotient
ring .
(c) Is an integral
domain?
(a)
(b) I will let ,
,
, and
stand for their respective cosets.
(c) Since (which is the zero
element in
), the
quotient ring
is not an
integral domain.
The best thing for being sad is to learn something. - Merlyn, in T. H. White's The Once and Future King
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