Homework 10
5.1.4. Claim: Let G be a group and H a subgroup of G. Then G acts on G/H, the set
of left cosets of H, by left multiplication, and this action is transitive.
Proof: First, if e G is the identity, and aH G/H, then e(AH) = (ea)H = aH,
so e fixes
Math 417 Exam 2 (Solutions) Prof. I.Kapovich November 11, 2011 Problem 1.[20 points] For each of the following statements indicate whether it is true or false. You DO NOT need to provide explanations for your answers in this problem. (1) If G is a group a
Math 417 Section C1 Exam 1 (Solutions) Prof. I. Kapovich September 30, 2011 Problem 1.[20 points] For each of the following statements indicate whether it is true or false. You DO NOT need to provide explanations for your answers in this problem. (1) The
Pre-class worksheet due Wednesday, November 16 (Solutions).
1. Consider the elements a = 8 + 18i and b = 2 + 3i in the ring Z[i] of Gaussian integers. Recall that Z[i] is a Euclidean ring with the norm N (x + iy) = x2 + y 2 where x, y Z. Note: There was a
Pre-class worksheet due Monday, November 14, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The ring C is a Euclidean domain.
(2) The ring Z is a Euclidean domain.
(3) The ring Z[x] is a Euclidean domain.
(4
Pre-class worksheet due Wednesday, November 9, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The ideal 5Z Z is a maximal ideal in Z.
(2) There does not exist a eld F such that F has four distinct ideals.
(3
Pre-class worksheet due Monday, November 7 (Solutions).
1. Consider the ring Z and the ideals I = 6Z, J = 4Z in Z.
Taking for granted that every ideal in Z has the form nZ for some n Z,
represent in this form each of the following ideals:
I + J, IJ, I 2 ,
Pre-class worksheet due Friday, November 4 (Solutions).
1. Consider the function : R[x] C given by the formula (f ) = f (i) for every polynomial f (x) R[x]. More specifically, for every f (x) = a0 + a1 x + + an xn R[x] we have (f ) := a0 + a1 i + a2 i2 +
Pre-class worksheet due Wednesday, November 2. (Solutions)
1. For each of the following statements indicate if it is true or false.
(1) If R, S are rings with 1 and : R S is a ring homomorphism then
(1) = 1.
(2) If R, S are rings with 1 and : R S is a rin
Pre-class worksheet due Monday, December 5, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The ring Q[x]/(x4 + 6x3 3x2 + 9x + 3) is a eld.
(2) The ring R[x]/(x2 + 2) is a eld.
(3) The ring C[x]/(x2 + 2) is a
Pre-class worksheet due Friday, December 2, 2011 (Solutions).
1. For each of the following statements indicate if it is true or false:
(1) The polynomial x4 + 6x3 3x2 + 9x + 3 is irreducible in Z[x].
(2) The polynomial x4 + 6x3 3x2 + 9x + 3 is reducible i
Pre-class worksheet due Wednesday, Aug 31 (solutions).
PRINT YOUR NAME:
1. Consider the set (R, +) as a group with respect to addition.
Compute the 3-d power of 5 R with respect to addition.
What is the order of the element 5 in (R, +)?
Answer:
The 3-d po
Pre-class worksheet due Monday, October 31 (Solutions).
1. Let G = S3 and consider the group ring R = ZG.
(a) Compute the following element in R:
2 (1 2) (1 2 3) (1 2 3) + 7(3 2 1) .
(b) Find an element r R such that r(1 (1 2 3) = (1 (1 2 3)r = 0.
(c) In
Pre-class worksheet due Wednesday, October 26 (Solutions).
1. For each of the following objects determine if it is a ring. If the object is
a ring, indicate if this ring is commutative and if it has an identity element.
(1) The set Z of integers, with the
Pre-class worksheet due Monday, October 24 (Solutions).
1.
(1) Let G be a group with |G| = 77. Find the number n7 of Sylow
7-subgroups of G.
(2) Let G be a nite p-group (where p 2 is a prime). Is it true that
every two Sylow p-subgroups of G are isomorphi
Pre-class worksheet due Wednesday, October 19 (Solutions).
PRINT YOUR NAME:
1. For each of the following statements indicate whether it is true or false.
(1) If G is a group with |G| = 100 then G is isomorphic to a subgroup
of S100 .
(2) For every group G
Pre-class worksheet due Monday, October 17 (solutions).
1. Consider the standard action of the symmetric group G = S5 on the
set X = cfw_1, 2, 3, 4, 5.
(1) Is this action transitive?
(2) Is this action faithful?
(3) Let G2 be the stabilizer in G = S5 of t