2
1
Let Fnbe the free group of rank n for n 2 and H a nite index subgroup of Fn.
Prove that H is a free group.
(You may assume that the free group of rank k is the free product of k copies of Z.)
If H has index i then state and prove a formula for the ran
2
(a) State and prove the spectral theorem for compact selfadjoint operators on a
1
Hilbert space.
(b) Dene the traceclass norm on the space of niterank operators. Prove that it is a
norm and that the dual space can be identied with B (H).
(c) Prove that
2
1
Show that a commutative Artinian ring is Noetherian.
State and prove Wedderburns theorem concerning the structure of a (not necessarily commutative) semisimple right Artinian ring.
2
Let R be a commutative Noetherian ring. Show that the power series R
2
1
a) Let W = x, y be the standard (dening) representation of sl(2). Let
U = W Sym3(W ).
(i) Decompose U into weight spaces, giving a basis for each weight space.
(ii) Draw the weight diagram for U , and identify which irreducible representations occur a
2
1
Suppose that A is a ring. Dene an Artinian A-module. What does it mean to say
that A is Artinian?
Prove carefully that if A is an Artinian ring and M is a nitely generated A-module
then M is an Artinian A-module.
Give an example of an Artinian ring A
2
Prove the Theorem of Jordan that a primitive permutation group of degree n
1
containing a 3-cycle contains the alternating group An.
Use this to classify the maximal subgroups of the symmetric group Sncontaining
a 3-cycle.
Give an example to show that f
2
1
Let k be an algebraically closed eld and let G be a nite group.
Prove that the number of isomorphism classes of simple kG-modules is equal to the
number of conjugacy classes of elements of G of order not divisible by the characteristic
of k.
Let H be
2
1
Let G be a nite group.
(a) Dene the Frattini subgroup (G) and Fitting subgroup F (G) of G. Prove that
F (G) exists.
(b) Show that (G) F (G).
(c) If G is soluble prove that F (G) contains its own centraliser.
(d) Let G be a nite group. The socle of G i
MATHEMATICAL TRIPOS
Part III
Thursday, 27 May, 2010 1:30 pm to 4:30 pm
PAPER 1
LIE GROUPS, LIE ALGEBRAS,
AND THEIR REPRESENTATIONS
Attempt no more than FOUR questions.
There are FIVE questions in total.
The questions carry equal weight.
STATIONERY REQUIRE
MATHEMATICAL TRIPOS
Part III
Monday 11 June 2007 9.00 to 12.00
PAPER 3
MODULAR REPRESENTATIONS OF FINITE GROUPS
Attempt THREE questions.
There are SIX questions in total.
The questions carry equal weight.
In this paper G is a nite group. In the usual nota
2
1
(i) Let G = g : g4= 1 and H = h : h10= 1 .
Let G0= g2G, H0= h5H and : G0= H0.
Prove that A := G H ( : G0= H0) has soluble word problem clearly stating
any facts about free products with amalgamated subgroups that you use.
Solve the word problem for A/
MATHEMATICAL TRIPOS
Part III
Tuesday 12 June 2001 9 to 12
PAPER 5
TOPICS IN REPRESENTATION THEORY
Candidates should attempt ALL the questions.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigil
MATHEMATICAL TRIPOS
Part III
Monday 4 June 2001 1.30 to 4.30
PAPER 3
TOPICS IN GROUP THEORY
Attempt THREE of the six questions,
at least ONE of which should be from Section B.
The questions carry equal weight.
You may not start to read the questions
print
MATHEMATICAL TRIPOS
Friday 2 June, 2006
Part III
9 to 12
PAPER 6
INTRODUCTION TO BANACH SPACES AND ALGEBRAS
Attempt FOUR questions.
There are SIX questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
Cover sheet
Treasury Tag
Script
2
1 (a) Describe Brauers characterisation of characters. (Dene all concepts needed which
are not obvious.)
(b) Let G be a nite group and A, B subsets of G such that G = A B cfw_1 is a
disjoint union. Suppose that if a A and b B then (o(a), o(b) = 1. Show
MATHEMATICAL TRIPOS
Part III
Friday 6 June 2008 1.30 to 4.30
PAPER 3
LIE ALGEBRAS AND REPRESENTATION THEORY
Attempt no more than THREE questions, which must include the rst question.
There are FIVE questions in total.
The 2nd - 5th questions carry equal w
2
1
(a) Let : F C be an embedding of elds and n
1 a natural number.
Is the induced homomorphism : GLn(F ) GLn(C), (aij) (aij), a smooth
representation? Explain your answer.
(b) Let G be an -group, V = Cc(G,C) the space of locally constant functions with
c
MATHEMATICAL TRIPOS
Part III
Monday, 8 June, 2009 1:30 pm to 4:30 pm
PAPER 4
CHARACTER THEORY OF FINITE GROUPS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Results from lectures and exercise sh
2
1
a) Let V be the standard (dening) representation of sl(2). Let
U = 2(Sym4V ).
i) Decompose U into weight spaces, giving a basis for each weight space.
ii) Draw the weight diagram for U , and identify which irreducible representations
occur as submodul
2
(i) Let g be a Lie algebra. Dene what it means for g to be (a) solvable and (b)
1
semisimple.
(ii) Let g = sln. Dene two bilinear forms (, ) and (, )adby (A, B) = tr(AB) and
(A, B)ad= tr(adA adB). Show (, ) = (, )adand compute .
(iii) Let g be the four
2
Dene the set Spec(A) for a ring A, and the Zariski topology on Spec(A). Show
1
that the Zariski topology is in fact a topology.
State Zorns Lemma, and use it to show that Spec(A) = whenever A = 0.
Show that Spec(A) denes a contravariant functor from the
2
1
(a) Dene SLq(2) and give its coalgebra and algebra structure explicitly.
(b) Dene the coaction of SLq(2) on kq[x, y] and compute explicitly
x2y.
(c) Explain what is meant by an R-point of kq[x, y]. What are the C points of
kq[x, y]? If R is the algebr
MATHEMATICAL TRIPOS
Part III
Friday 31 May 2002 1.30 to 4.30
PAPER 3
CONSTRUCTIVE GALOIS THEORY
Attempt THREE questions
There are ve questions in total
The questions carry equal weight
You may not start to read the questions
printed on the subsequent page
2
Section A
1
map
Let G be a Lie group and LG its associated Lie algebra. Dene the exponential
exp : LG G.
Show (a) that exp is a local dieomorphism at the origin O in LG, and (b) that
exp is a homomorphism if and only if G is abelian. Deduce that if G is
MATHEMATICAL TRIPOS
Part III
Friday, 29 May, 2009 1:30 pm to 4:30 pm
PAPER 6
FINITE DIMENSIONAL LIE ALGEBRAS
AND THEIR REPRESENTATIONS
Attempt ALL questions.
There are FIVE questions in total.
Question THREE carries the most weight.
All Lie algebras are o
MATHEMATICAL TRIPOS
Part III
Thursday 7 June 2001 1.30 to 4.30
PAPER 6
LINEAR ANALYSIS
Answer FOUR questions. The questions carry equal weight.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigi
MATHEMATICAL TRIPOS
Part III
Friday 6 June 2003 9 to 12
PAPER 5
GEOMETRIC GROUP THEORY
Attempt THREE questions.
There are ve questions in total.
The questions carry equal weight.
Notation and Conventions Let A be a nite set, A is the set of all nite words
MATHEMATICAL TRIPOS
Part III
Monday 3 June 2002 9 to 12
PAPER 5
COMPLEX ANALYSIS
Attempt FOUR questions
There are six questions in total
The questions carry equal weight
You may not start to read the questions
printed on the subsequent pages until
instruc
2
1
Let k be a eld and G be the group of upper unitriangular integral 3 3 matrices
G=
1 a b
0 1 c : a, b, c Z
0 0 1
(i) By proving an appropriate version of Hilberts Basis Theorem, show that kG is
both left and right Noetherian.
(ii) Prove that if r an
MATHEMATICAL TRIPOS
Part III
Friday 28 May, 2004 1.30 to 4.30
PAPER 6
INTRODUCTION TO FUNCTIONAL ANALYSIS
Attempt THREE questions.
There are four questions in total.
The questions carry equal weight.
You may not start to read the questions
printed on the