A look at a few Tripos questions X

Since time is short, I am going to discuss a couple of Groups questions but in slightly less detail than I have been giving up to now: instead of working through the questions completely, I’ll try to zero in on the most important points. Because there wasn’t a separate Groups course until 2008, I am taking my questions from that year.

5E. For a normal subgroup of a group , explain carefully how to make the set of (left) cosets of into a group.

For a subgroup of a group , show that the following are equivalent:

(i) is a normal subgroup of ;

(ii) there exist a group and a homomorphism such that is the kernel of .

Let be a finite group that has a proper subgroup of index (in other words, ). Show that if , then cannot be simple. [Hint: Let act on the set of left cosets of by left multiplication.]

A look at a few Tripos questions IX

Exam day approaches, so I’ve decided to prioritize. Instead of doing question 7C, which isn’t all that interesting (in the sense that it doesn’t give me much scope to emphasize principles of more general use in exams), I’m going to skip it, and instead, if I get time, go through a groups question or two. But first I will do the final Numbers and Sets question because it involves something a lot of people dislike: the inclusion-exclusion principle. It’s worth getting comfortable with this, because it comes up year after year (either in Numbers and Sets or in Probability). You may think that applying the principle requires some ingenuity. The aim of this post is to convince you that it can be done on autopilot.

8C. Let be a finite set with elements. How many functions are there from to ? How many relations are there on ?

Show that the number of relations on such that, for each , there exists at least one with , is .

Using the inclusion-exclusion principle or otherwise, deduce that the number of such relations for which, in addition, for each , there exists at least one with , is

.

A look at a few Tripos questions VIII

Now for a question on modular arithmetic. As with countability, there is a very high chance of a question on this topic. [Added after the post was written: as usual I wrote down my thoughts about this question as I had them, and I didn't spot the best approach to part (ii) of the question until after I had come up with some less good approaches. So my recommendations evolve through the post, with some of the later ones superseding some of the earlier ones.]

6C. (i) Prove Wilson’s theorem: if is prime then (mod ).

Deduce that if (mod 4) then

(mod ).

(ii) Suppose that is a prime of the form . Show that if (mod ) then (mod ).

(iii) Deduce that if is an odd prime, then the congruence

(mod )

has exactly two solutions (modulo ) if (mod 4), and none otherwise.

A look at a few Tripos questions VII

The obligatory question on countability/uncountability.

5C. Define what is meant by the term countable. Show directly from your definition that if is countable, then so is any subset of .

Show that is countable. Hence or otherwise, show that a countable union of countable sets is countable. Show also that for any , is countable.

A function is periodic if there exists a positive integer such that, for every , . Show that the set of periodic functions is countable.

A look at a few Tripos questions VI

I’m now going to turn to the Numbers and Sets questions from the same year, 2003. I’ve lost count of the number of times I’ve heard people say that the course is quite easy but the questions on the examples sheets and exams are very hard and “not very closely related to the course”. There is a grain of truth in that: the new concepts you have to grasp in Numbers and Sets are not as difficult as the new concepts you have to grasp in most of the other courses, so in order to give enough substance to Tripos questions the examiners are almost forced to put in a significant problem-solving element. However, certain styles of problem occur quite regularly, so it’s good to get a bit of practice. And perhaps a detailed discussion of the 2003 questions will be helpful as well. As I did with Analysis I, I’ll start with a post on the Section I questions and then I’ll have separate posts for each of the four Section II questions. The paper, by the way, is Paper 4.

A look at a few Tripos questions V

Here is the final analysis question from 2003.

12C. State carefully the formula for integration by parts for functions of a real variable.

Let be infinitely differentiable. Prove that for all and for all ,

.

By considering the function at , or otherwise, prove that the series

converges to .

A look at a few Tripos questions IV

This post belongs to a series that began here. Next up is a question about integration.

11B. Let be continuous. Define the integral . (You are not asked to prove existence.)

Suppose that are real numbers such that for all . Stating clearly any properties of the integral that you require, show that

.

The function is continuous and non-negative. Show that

.

Now let be continuous on . By suitable choice of show that

,

and by making an appropriate change of variable, or otherwise, show that

.

A look at a few Tripos questions III

Here’s another one.

10F. State without proof the Integral Comparison Test for the convergence of a series of non-negative terms.

Determine for which positive real numbers the series converges.

In each of the following cases determine whether the series is convergent or divergent:

(i) ,

(ii) ,

(iii) .

A look at a few Tripos questions II

This is the second in a series of posts that started here. In the first post I explained what I’m up to. Now let me just continue with some more questions. I’m now on to the harder Section II questions. Here’s the first one I want to look at. Even though it makes the posts shortish, I think I’m going to stick to one long question per post.

---

9F. Prove the Axiom of Archimedes.

Let be a real number in and let be positive integers. Show that the limit

exists, and that its value depends on whether is rational or irrational.

[You may assume standard properties of the cosine function provided they are clearly stated.]

A look at a few Tripos questions I

When I was a mathematics undergraduate, I became aware of a huge cultural difference between mathematicians and engineers. That sounds like the beginning of a joke you’ve heard twenty times already, but it isn’t. The difference was that when mathematicians were set questions, they were expected to work out how to solve them, and if they couldn’t do so then it was too bad — the best they could do about it was ask their supervisors. But engineers had model answers for everything, available with the latest technology, which in those days was microfiche. In case you have no idea what I’m talking about, the answers were reduced in size by a factor of about five in each direction and printed on to some kind of transparent plastic that you could look at through a magnifying machine. There were a couple of the machines in our college library, and they were nearly always in use.

Model answers have always seemed to me to be a bad idea in mathematics, because it is hard to learn how to think for yourself when you are given the answers to all the problems you tackle. So it might seem a bit odd that in this post I’m going to attempt to help people preparing for Part IA of the Cambridge Mathematical Tripos by providing some model answers.