MATH 254, AUTUMN 2015
HOMEWORK 8
DUE WEDNESDAY, DECEMBER 2
1. Suppose that F is a eld and a = 0 is a zero of the polynomial
f (x) = a0 + a1 x + . . . + an xn F [x].
Prove that 1/a is a zero of an + an
Homework Two
Problem I.
(1) Show that the set cfw_i, 1, i, 1 is a group under complex multiplication.
(2) Let r be the symmetry of a square card given by rotating it
through an angle of /2. Show that
Homework Six
Problem I Suppose G acts on a set S. Prove that the orbits of the
action partition S into equivalence classes.
Problem II Let p be a prime number and let G be a finite group
of order pr a
Homework One
Problem I. Read Chapter 0 of the textbook (A First Course in
Abstract Algebra by Fraleigh) and do exercises 29 through 32 of this
section.
Problem II. Let R be an equivalence relation on
Homework Five (part 1 of 2)
Problem I.
(1) Prove that if N G is a subgroup then the following two
properties are equivalent:
(a) for every g G we have gN = N g, and
(b) for every g G we have gN g 1 =
BASIC ALGEBRA 1, Math 25400
Sec. 11 and Sec. 31
Homework 1
(due Friday October 10th)
Section 0: 18
Section 2: 8, 24, 26, 36
Section 4: 14, 19, 34
1. Let X = cfw_1, 2, 3 and denote by S3 the symmetric
BASIC ALGEBRA 1, Math 25400
Sec. 11 and Sec. 31
Homework 2
(due Friday October 17th)
Section 4: 20
Section 5: 12, 20, 26, 32, 43, 52
Section 6: 18
Recall: For any positive integer n, the group Sn cons
Homework Three
Problem I.
(1) Let Q
subgroup of
GL2 C generated
by the matrices
8 be the
0 1
0 i
and B =
where i2 = 1. Show
A =
1 0
i 0
that Q8 is a non-abelian group of order 8. (Hint: observe
Homework Five (part 2 of 2)
Problem I.
(1) Let G be a finite group. Prove that if : G G0 is a homomorphism, then
|G| = |Ker()| |(G)|.
(2) Let G and G0 be finite groups. Prove that if |G| and |G0 | are
Homework Seven (worth double)
Problem I.
Give a counterexample to Cauchys Theorem if we drop the assumption that p is prime. That is, find a group G and a number k dividing
the order of G such that th
MATH 254, AUTUMN 2015
HOMEWORK 7
DUE WEDNESDAY, NOVEMBER 25
1. In each of the following, describe all solutions of the given congruence
(a) 39x 52 (mod 130)
(b) 39x 125 (mod 9)
(c) 22x 5 (mod 15)
(d)
MATH 254, AUTUMN 2015
HOMEWORK 1
DUE FRIDAY, OCTOBER 9
1. For any set S, let P(S) denote the collection of subsets of S. For instance, if
S = cfw_1, 2, 3, 4
then cfw_1, 3, 4 P(S), is an element of P(S
MATH 254, AUTUMN 2015
HOMEWORK 6
DUE FRIDAY, NOVEMBER 20
1. Consider the ring M2 (Z2 ) of 2 2 matrices with coecients in Z2 .
(a) Determine the number of elements of M2 (Z2 ).
(b) Find all units of th
MATH 254, AUTUMN 2015
HOMEWORK 3
DUE FRIDAY, OCTOBER 23
1. Let G be a cyclic group with generator a, and let G be a group isomorphic to G.
If : G G is an isomorphism, show that for every x G, (x) is c
MATH 254, AUTUMN 2015
HOMEWORK 5
DUE FRIDAY, NOVEMBER 13
1. Let : G G be a group homomorphism, and let N be a normal subgroup of
G . Prove that 1 [N ] is a normal subgroup of G.
2. In each of the foll
MATH 254, AUTUMN 2015
HOMEWORK 4
DUE FRIDAY, NOVEMBER 6
1. Let G be a group of order pq, where p and q are prime numbers. Prove that every
proper subgroup of G is cyclic.
2. Let G be a group and H G a
MATH 254, AUTUMN 2015
HOMEWORK 2
DUE FRIDAY, OCTOBER 16
1. Let B be a set of binary structures, i.e. an element of B is a binary structure
S, . Prove that isomorphism S,
S , denes an equivalence rela
Homework Four
Problem I.
Let : G G0 be a surjective group homomorphism.
(1) Assume that G is cyclic. Prove that G0 is cyclic.
(2) Assume that G is abelian. Prove that G0 is abelian.
(3) Find counterex
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