MATH30300: Group Theory
Homework 4: Solutions
1. (a) Recall that if R and if w R2 , we let R,w denote rotation
about w through an angle of radians. As usual, R means R,0 .
Let v R2 and let R be a nontrivial rotation. Prove that Tv R
is a rotation. Specica
MATH30300: Group Theory
Homework 5
Please submit answers to the SIX questions marked with at the
beginning of the tutorial on Thursday 24th November.
1 . Express each of the following permutations as a product of disjoint
cycles:
(a) The permutation S8 gi
MATH30300: Group Theory
Homework 4
Please submit answers to questions marked with at the beginning of the tutorial on Thursday 10th November.
1. (a) Recall that if R and if w R2 , we let R,w denote rotation
about w through an angle of radians. As usual, R
MATH30300: Group Theory
Homework 5
Please submit answers to the SIX questions marked with at the
beginning of the tutorial on Thursday 24th November.
1 . Express each of the following permutations as a product of disjoint
cycles:
(a) The permutation S8 gi
MATH30300: Group Theory
Homework 1: Solutions
1. For each of the following functions, decide whether it is injective,
surjective or bijective. Where it is injective, write down a left inverse,
and where surjective a right inverse.
(a) f : R R, f (x) = x3
MATH30300: Group Theory
Homework 2
1. (a) If v is a vector in R2 , let Tv be translation by v; i.e. Tv is the
isometry R2 R2 given by Tv (w) = w + v. Draw a picture
to show that the isometry Tv R Tv : R2 R2 is rotation
anticlockwise through an angle of ra
MATH30300: Group Theory
Homework 2
1. (a) If v is a vector in R2 , let Tv be translation by v; i.e. Tv is the
isometry R2 R2 given by Tv (w) = w + v. Draw a picture
to show that the isometry Tv R Tv : R2 R2 is rotation
anticlockwise through an angle of ra
MATH30300: Group Theory
Homework 3: Solutions
1. Let G be a group with identity element e. Show that e is the unique
identity element.
Solution: Suppose e is another identity. Then we have e g = g e =
g for all g G. In particular, e e = e. However, from t
MATH30300: Group Theory
Homework 1
We use the following notation below ( is a real number):
R =
cos sin
sin cos
,
S =
cos sin
sin cos
.
R is the matrix representing rotation about 0 through radians, and S is
the matrix of reection through the line thr
MATH30300: Group Theory
Homework 3
1. Let G be a group with identity element e. Show that e is the unique
identity element.
2. Let U = Z2 be the subset of R2 consisting of all points with integer
coordinates; i.e. all points of the form (m, n) with m, n Z