Math 5125
Monday, October 10
Sixth Homework Solutions
1. Let F denote the free group on x, y, let a be a generator for K , and dene a homomorphism : F G by x = a and y = b (where a = (a, 1) and b = (1, b).
Note that is onto, because G is generated by a an
Math 5125
Monday, September 5
Second Homework Solutions
1. Let G be a group with distinct normal subgroups A, B such that |G : A| = |G : B| = 2.
(a) Prove that AB = G.
(b) Prove that A/A B and B/A B are distinct normal subgroups of G/A B of order
2 and in
Math 5125
Monday, November 14
Tenth Homework Solutions
1. (a) Both sides have degree n and both sides have roots cfw_ i | i = 1, . . . , n. Since both
sides have constant term, it follows that both sides are equal.
(b) More generally in C[x, y], we have 1
Math 5125
Monday, November 21
Eleventh Homework Solutions
1. Section 10.3, Exercise 7 on page 356. Let R be a ring with a 1, let M be a left Rmodule, and let N be a submodule of M . Prove that if both M /N and N are nitely
generated, then so is M .
Since
Math 5125
Wednesday, December 7
Twelfth Homework Solutions
1. Suppose R is a commutative ring and let I , J be ideals of R.
(a) Show that there is a surjective R-module homomorphism from I R J to IJ mapping i j to i j.
(b) Give an example to show that the
Math 5125
Monday, August 22
August 22, Ungraded Homework
Exercise 3.1.36 on page 89 Prove that if G/ Z(G) is cyclic, then G is abelian.
Write Z = Z(G). If G/Z is cyclic, then G/Z = xZ for some x G. This means that the left
cosets of Z in G are of the form
Math 5125
Monday, October 3
Monday, October 3
Exercise 6.3.7 on page 220 Prove that the following is a presentation for the quaternion
group of order 8:
Q8 = a, b | a2 = b2 , a1 ba = b1 .
Q8 is a group with 8 elements, namely cfw_1, 1, i, j, k with identi
Math 5125
Wednesday, November 30
Sample Final Problems since Second Test
1. Let F be a eld with characteristic = 2.
(a) Prove that F [x]/(x 1) F [x]/(x + 1) as rings.
=
(b) Prove that F [x]/(x 1) F [x]/(x + 1) as F [x]-modules.
(c) Is F [x]/(x 1) F [x]/(x
Math 5125
Friday, November 4
November 4, Ungraded Homework
Remarks If M is an R-module, then 0m = 0 = r0, and (r)m = rm for all m M and
r R. We shall use these elementary properties without comment.
Exercise 10.1.3 on page 343 Assume that rm = 0 for some
Math 5125
Monday, December 5
Solutions to Sample Final Problems
1. (a) Dene a ring homomorphism : F [x] F by ( f ) = f for f F and (x) = 1.
Then is onto and ker = cfw_ p F [x] | p(1) = 0. Clearly x 1 ker , and by
the factor theorem, ker (x 1). Therefore k
Math 5125
Monday, August 22
August 22, Ungraded Homework
Exercise 3.1.36 on page 89 Prove that if G/ Z(G) is cyclic, then G is abelian.
Write Z = Z(G). If G/Z is cyclic, then G/Z = xZ for some x G. This means that the left
cosets of Z in G are of the form
Math 5125
Wednesday, August 24
August 24, Ungraded Homework
Exercise 3.3.3 on page 101 Prove that if H is a normal subgroup of G of prime index p,
then for all K G either
(i) K H or
(ii) G = HK and |K : K H | = p.
We know that HK G, because H , K G and on
Math 5125
Monday, October 31
Ninth Homework Solutions
1. Let R = Z[ n] where n is a square free integer greater than 3.
(a) Prove that 2, n and 1 + n are irreducibles in R.
(b) Prove that R is not a UFD.
(c) Give an explicit ideal in R that is not princip
Math 5125
Monday, October 24
Eighth Homework Solutions
1. (a) Let a, b S and suppose S is not multiplicative, i.e. there exists 0 = r R such
that abr = 0. Since b is a nonzerodivisor, we see that br = 0. But then a(br) = 0,
which contradicts the hypothesi
Math 5125
Monday, September 12
Sample First Test no. 1. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let G be a group with exactly three elements of order 2. Prove that G is not simple.
(17 points)
2. Prove that a group of order 1683
Math 5125
Wednesday, September 13
Sample First Test no. 2. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let G be a group with normal subgroups of index 3 and 5. Prove that there exists a
group homomorphism of G onto Z/15Z.
(16 points
Math 5125
Wednesday, September 21
Sample First Test no. 3. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let p be a prime, let n be a positive integer, and let G be a noncyclic group of order pn .
Prove that there exists K G such that
Math 5125
Monday, October 24
Sample Second Test no. 1. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let R be a commutative ring and let I denote the set of nilpotent elements of R (that
is, cfw_r R | rn = 0 for some positive integer
Math 5125
Wednesday, September 28
First Test. Answer All Problems.
Please Give Explanations For Your Answers
1. Let H be a subgroup of the nite group G, and let p be a prime. Prove that two
distinct Sylow p-subgroups of H cannot be contained in the same p
Math 5125
Wednesday, November 2
Second Test. Answer All Problems.
Please Give Explanations For Your Answers
1. Suppose R = 0 is a commutative ring with a 1. Prove that the set of prime ideals in R
has a minimal element with respect to inclusion.
(17 point
Math 5125
Monday, August 29
First Homework Solutions
1. Section 3.2, Exercise 8 on page 95. Prove that if H and K are nite subgroups of the
group G whose orders are relatively prime, then H K = 1.
Since H K is a subgroup of the nite group H , we see that
Math 5125
Monday, September 12
Third Homework Solutions
1. The action of G on G/H (the left cosets of H in G) yields a homomorphism : G
Sn , and by hypothesis (G) An . Note that (g) has order 1 or 2, because g has order
2. Since (g) An and n 4, we see th
Math 5125
Monday, September 19
Fourth Homework Solutions
1. We have a homomorphism : G Aut(A) determined by ( g)(a) = gag1 . If g A,
then gag1 = a for all a A because A is abelian, hence (g) is the identity and we
deduce that g ker . Therefore A ker . Sin
Math 5125
Monday, September 26
Fifth Homework Solutions
1. Certainly H G. Now let Aut(G). Since K char G, we see that K = K , so the
restriction of to K is an automorphism of K . Now H char K , hence H = H and
we deduce that H = H . This proves that H cha
Math 5125
Monday, October 17
Seventh Homework Solutions
1. Note that nZ mZ because m n. By the ideal correspondence theorem we now obtain
Z/nZ
= Z/mZ.
mZ/nZ
However the ideals mZ/nZ and m(Z/nZ) of Z/nZ are equal, because they both consist of the cosets m
Math 5125
Friday, August 26
August 26, Ungraded Homework
Exercise 4.1.4 on page 116 Let S3 act on the set of ordered pairs: cfw_(i, j) | 1 i, j 3
by (i, j) = ( (i), ( j). Find the orbits of S3 on . For each S3 nd the cycle
decomposition of under this acti
Math 5125
Monday, August 29
August 29, Ungraded Homework
Exercise 4.2.9 on page 130 Show that | CenSn (1 2)(3 4)| = 8 (n 4)! for all n 4.
Determine the elements in this centralizer explicitly.
|K (1 2)(3 4)| = n(n 1)(n 2)(n 3)/23 (because we can rotate ea
Math 5125
Friday, September 6
Third Homework
Due 9:05 a.m., Friday September 13
1. Let n N, let G be a group with a subgroup H of index n, and let g G be an element
of order 2. Suppose 4 n, and that whenever : G Sn is a homomorphism, then
(G) An . Prove
Math 5125
Monday, September 16
Third Homework Solutions
1. The action of G on G/H (the left cosets of H in G) yields a homomorphism : G
Sn , and by hypothesis (G) An . Note that (g) has order 1 or 2, because g has order
2. Since (g) An and n 4, we see th
Math 5125
Monday, August 26
First Homework
Due 9:05 a.m., Friday August 30
1. Section 3.2, Exercise 8 on page 95.
(2 points)
2. Let A be a normal abelian subgroup of the group G and let H G. Suppose AH = G.
Prove that A H G. Hint: if g G, write g = ah and