AssignmerAS Solutions
' 1. The general method of solution is to express the permutations as
products of disjoint cycles. We then can determine the order as the lcm
of the lengths of the cycles. Let us use the notation L0, for a perfect in
shufe and 00, fo

MATH 3031 : Abstract Algebra
Review for test 1
The midterm test on Monday, Oct. 17 will be on material covered in class
up to and including the class on Oct. 13. This material is contained in
sections 2,3,4,5,6,8, 9, 10 of the textbook. Some practice prob

C)
Assignmentl Solutions
1. (a) The group (Z/12Z >< Z/18Z) is of order 12- 18 hence the quotient
group will be of; order 12 - 18 / m where m is the order of the element
(4,3) in the product. ~m is the smallest positive integer for which 4m
is divisible by

MATH 3031
Abstract Algebra
Assignment 5
1. Perfect shues are permutations of the form
(
1 2 3 . n n + 1 n + 2 .
2 4 6 . . . 2n
1
3
.
or
(
1 2 3
1 3 5
.
.
n
n+1
2n 1
2
n+2
4
2n
2n 1
.
.
2n
2n
)
)
The rst is called a perfect in shue and the second a perfect

MATH 3031
Abstract Algebra
Assignment 8
1. Use Lagranges theorem applied to the multiplicative group (Z/pZ)
of nonzero elements of Z/pZ with multiplication as group operation to
prove Fermats little theorem: that if p is a prime and a is not divisible
by

MATH 3031
Abstract Algebra
Assignment 6
1.(a) Find the order of the quotient group
(Z/12Z Z/18Z)/ < (4, 3) >
(b) Find the order of the element (3, 1)+ < (1, 1) > in the quotient group
(Z/4Z Z/4Z)/ < (1, 1) >.
2. Let n be a positive integer and let (Z/nZ)

MATH 3031
Abstract Algebra
Assignment 4
1. Find the gcd of 58 and 88 using the Euclidean algorithm. Find
integers x and y such that
x 58 + y 88 = gcd(58, 88)
(
2. Let
=
(
and
=
1 2
3 1
3
4
4
5
5
6
6
2
1
2
3
1
4
3
5
6
6
5
2
4
)
)
be elements of S6 . Calcul

MATH 3031
Abstract Algebra
Assignment 3
1.Find the number of elements in the cyclic subgroup of C (the nonzero
complex numbers with multiplication as binary operation) generated by
each of the following elements:
(a) i
(b) (1 + i)/ 2
(c) 1 + i
2.How many

MATH 3031
Abstract Algebra
Assignment 7
1. The centre of a group, G, is the set
Z(G) = cfw_z G|zg = gz for all g G
of elements of G which commute with all other elements. Show that
Z(G) is a normal subgroup of G. Find Z(S3 ), Z(D4 ) and Z(S3 D4 ).
2.(a) S