Groups and Symmetry Problem Set 2
1
1 1
Q1: Note that R (x) = x. Give expressions for Ra, (x), Ra, (x) and Ra, Rb, Ra, Rb, (x).
1 1
Check that your last answer is consistent with the formula in the no
Groups and Symmetry Problem Set 2 Solutions
Q1: First, we have Ra, = Ta R Ta , so Ra, (x) = a + R (x a) = a (x a) = 2a x. If
1
1
y = 2a x then x = 2a y; this shows that Ra, (y) = 2a y and thus Ra, = R
Groups and Symmetry Problem Set 5 Solutions
Q1: Put G = Dir(X), which is a nite subgroup of SO3 . The normalisations of the midpoints of
the edges are poles of degree 2, the normalisations of the cent
Groups and Symmetry Problem Set 1
Please hand in problems 1 and 5 on Tuesday 21st October.
Q1: Find Symm(X) and Dir(X) when X R2 is
(a) The unit disc centred at the origin
(b) The isosceles triangle w
Groups and Symmetry Problem Set 3
Please hand in questions 1 and 4 on Tuesday November 4th.
Q1: Let X be the wallpaper pattern shown below. We have seen that I(X) = Tu , Tv , R/3 , S0 ,
where u = (1,
Groups and Symmetry Problem Set 3 Solutions
Q1: We have
GL,u/2 (0, 0) = u/2 + SL (0, 0) = (1/2, 0) + (0, 3/2) = v,
so the map f = Tv GL,u/2 = Tu/2v SL satises f (0, 0) = (0, 0). As L is parallel to th
Groups and Symmetry Problem Set 1 Solutions
Q1:
(a) Here Symm(X) = O2 and Dir(X) = SO2 . This just means that the unit disc is invariant
under any rotation about the origin, and under any reection acr
Groups and Symmetry Problem Set 4
Please hand in questions 1 and 4 on Tuesday 18th November.
Q1: Let g : R3 R3 be the map g(x, y, z) = (y, z, x).
(a) Prove that g O3 .
(b) Find a unit vector u with g(
Groups and Symmetry Problem Set 6 Solutions
Q1: The identity element xes all of X, so it has more than one xed point. The orbit counting
theorem says that the average number of xed points is the numbe
GROUPS AND SYMMETRY ABSTRACT GROUP THEORY
N. P. STRICKLAND
1. The Sylow theorems
Let G be a nite group of order n, say. Lagranges theorem says that if H is a subgroup of G
and |H| = d, then d is a div
Groups and Symmetry Examinable Proofs
All denitions are examinable. All results and examples in the notes and problem sheets may
provide insight that you will need to solve problems in the exam. The r
Groups and Symmetry Problem Set 6
Q1: Let G be a nite group, and let X be a set with an action of G. Suppose that there is
precisely one orbit, and that |X| > 1. Use the orbit counting theorem to show
Groups and Symmetry Problem Set 4 Solutions
Q1:
(a) It is clear that g is linear, and we have
g(x, y, z)
2
= y 2 + z 2 + x2 = x2 + y 2 + z 2 = (x, y, z) 2 ,
so g preserves lengths. Thus g O3 .
(b) Vis
GROUPS AND SYMMETRY
N. P. STRICKLAND
1. Symmetry groups in Rn
1.1. General linear groups. We write Mn or Mn (R) for the set of n n matrices over the real
numbers. Recall that an nn matrix A is inverti
Groups and Symmetry Problem Set 5
Please hand in questions 1 and 4 on Tuesday 25th November.
Q1: The following diagram shows a cuboctahedron X in R3 , centred at the origin. Its faces are
squares and
f x g g Xt Xm g tgu H T P r T P m tg T P r T P `p ` `
1 Iwea|WUhqX [email protected]"Xvt
mtg h u tg X o H g T P r T P mtg T P r T P
vIwXssvpqu % [email protected][email protected]@SuvS
`g ` u h x g g h u tgu X o tX `
Algebraic Topology Problem Set 3
Please hand in questions 2 and 3 on Tuesday 11th March.
Q1:
In this problem we give an alternative way of understanding the map q : S n RP n .
Suppose that A Mn+1 R an
Algebraic Topology Problem Set 4
No questions to be handed in this week.
Q1: Put X = cfw_(x, y) R2 | y 2 = x3 x. (This is an example of an elliptic curve; such curves
are important in number theory an
Algebraic Topology Problem Set 1
Please hand in questions 1 and 3 on Tuesday 25th February.
Q1:
Which of the following rules gives a well-dened, continuous function f : X Y ? Justify your
answer with
Algebraic Topology Problem Set 7
Please hand in questions 2, 3 and 4 on Tuesday 6th May.
Q1: Let X be a metric space, and let f, g : X S 1 be continuous maps. Suppose that f is
linearly homotopic to g
Algebraic Topology Problem Set 6
Please hand in questions 1,2 and 3 on Tuesday April 1st.
Q1: Let X and Y be spaces such that 0 X and 0 Y are nite sets. Suppose that f : X Y and
g : Y X are continuous
GUIDANCE ON THE EXAM
.
1. Examples
This list covers most of the examples that you should know about. The exam may ask you to
prove that two spaces are (or are not) homeomorphic (or homotopy equivalent
PMA333 EXAMINATION SPRING 2001 SOLUTIONS
N. P. STRICKLAND
(1)
(2)
(i) A metric space X is compact if for every sequence (xn ) in X there is a subsequence
(xnk ) and a point x X such that xnk x.
(ii) L
Algebraic Topology Problem Set 5
Please hand in questions 1,2 and 3 on Tuesday 25th March.
Q1: Let f : X R \ cfw_0 be a continuous map. Show that f is homotopic to a map g such that
g(x)2 = 1 for all
m
A
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z n c j dl b
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z n c d b x dl b
Sa~a7pQ5ppCH
h c dh dr vg u cfw_ z z y i c gh v ps ss dh v y
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