MA20216 Algebra 1A (Winter ’13)
Tutorials
I give tutorials for Groups F5 and F6 at 12:15 on Thursdays in 1WN 3.10, for Groups G1 and G2 at 14:15 on Fridays in 8W 2.24, and for Groups A5 and A6 at 15:15 on Fridays in 1W 3.6. Both groups should hand in work to my folders in the pigeonholes for the module on the first floor of 4W.
Course Website
The webpage for this module can be found here.
Week 1
I was in France this week, so tutorials were covered by Elvijs Sarkans and Acyr Locatelli.
Week 2
The tutorials began this week with me repeating things you were probably already told, about how to do well on the course, use tutorials properly, and write clear mathematical arguments. We also discussed how to understand the handshaking problem on the first problem sheet using graphs. Some of you are confused about what “injective” and “surjective” mean, partly because they are in a way mirror to the definition of a function. Consider a function \(f\) from a set \(A\) to a set \(B\). This means that to every element \(a\) of \(A\), we associate a unique element of \(B\), which we call \(f(a)\). Swapping \(A\) and \(B\) in the requirement that every point of the domain should have an image in the codomain, we get the condition that all points in the codomain should be the image of something in the domain—functions that satisfy this are called surjective. Swapping the roles of \(A\) and \(B\) in the uniqueness condition, we get the condition that every point of \(B\) should be the image of at most one point of \(A\) (but possibly of none). Functions that satisfy this condition are called injective. If we take the mirror of the entire definition of a function at once, we find that every element of \(B\) should be the image of a unique element of \(A\), which is the definition of a bijection. (The fact that the definition of a bijection is the mirror image of the definition of a function is not an accident—bijections are precisely the functions which have inverses). These terms are difficult to understand and remember at first, so you should practice problems and examples related to them. A possible mnemonic; injective maps put \(A\) inside \(B\) without squashing it, whereas surjective maps put an element of \(A\) sur (French, meaning on top of) each element of \(B\).
Week 3
This week I am in Oxford on Thursday and Cardiff on Friday, so tutorials will be covered again, hopefully by the same people as in week 1. (Sorry—all being well, this should be the last time I miss a tutorial this semester).
Week 4
The theme this week was of proofs and counterexamples. As a rule of thumb, if your statement starts with “for all”, then to show that it is true you need to give a general proof, but to show that it is false, it suffices to produce a single explicit counterexample. Similarly, if your statement starts with “there exists”, then you can show it is true by producing an explicit example, but to show it is false requires a general proof. It is not a coincidence that these rules are opposite to one another; “for all” statements are the converses to “there exists” statements, and vice versa. There are some occasions in which it is easier to use a devious argument to show that something must exist without finding it explicitly. One example is proving the existence of a continuous function that is nowhere differentiable—such things exist, but are essentially impossible to write down explicitly, although there are constructions involving limits. As an example, that was confusing to some of you, the first question on the new sheet asks you to show that all primes bigger than \(3\) are of the form \(6m+1\) or \(6m-1\) for some natural number \(m\). Note that this requires a general argument, as you need to show something for every element of the (infinite) set of primes. It is not sufficient (or necessary) to produce examples of primes in the required forms.
Week 5
Despite my hope in Week 3, the Thursday tutorial this week was cancelled due to my support for the UCU strike action; see here. Sorry for the inconvenience. Most of you are now getting to the point where you are familiar with the general mechanics of proving things, and are using these skills effectively on the new problems, which is good to see. As such, no particular concerns stood out at the tutorial. The material this week on addition and multiplication tables for \(\mathbb{Z}_n\) prepares the ground for the general definitions of groups and rings, which will appear soon. As an extra primer, you could think about similarities (and differences) between addition and multiplication of integers, and addition and multiplication of matrices.
Week 6
Some of you were confused about modular arithmetic. In simplest terms (which will allow you to start doing it, but ideally you should strive for a more general understanding) when we do arithmetic modulo \(n\), we are allowed to replace any integer we see by a different integer that gives the same remainder on division by \(n\). So working modulo \(5\), we could replace \(4\) by \(-1\), for example. Here, by arithmetic, we mean addition (and subtraction) and multiplication; we can’t modify numbers in exponents. (In this context, these aren’t really integers anyway, but instead refer to performing an operation repeatedly some number of times). As an example, we can compute \(62 \times 64 \bmod 63\) by replacing \(62\) by \(-1\) and \(64\) by \(1\), to get \(-1 \times 1 = -1 \equiv 62 \bmod 63\). Some of the results from your lectures amount to proving that we can perform these replacements either before or after carrying out the addition or multiplication, without affecting the result. One further useful note; the proof of the Chinese remainder theorem gives you a general method for constructing solutions to simultaneous congruence equations (such as “\(x \equiv 5 \bmod 8\) and \(x \equiv 3 \bmod 5\)”). However, when working with small numbers, it can be quicker to use other methods.
Week 7
The question that everybody asked was what \(\overline{\zeta}\) means; this is complex conjugation. \[\overline{a+\mathrm{i}b}=a-\mathrm{i}b\] It is very useful for a number of this week’s questions to be able to prove that a polynomial of degree \(2\) or \(3\) in \(K[X]\) (where \(K\) is a field) is irreducible if and only if it has no roots in \(K\). This is because if such a polynomial were reducible, it would necessarily have a linear factor, and hence a root in \(K\). However, it is important to remember that this does not work for polynomials of higher degree. For example, the polynomial \(X^4+1\) in \(\mathbb{R}[X]\) has no real roots, but factors as: \[X^4+1=(X^2+\sqrt{2}X+1)(X^2-\sqrt{2}X+1)\]
Week 8
Remember not to accidentally use “proof by notation”. For example, you can’t prove that \((x^{-1})^n=(x^n)^{-1}\), for \(x\) an element of a group, by writing \((x^{-1})^n=x^{-n}=(x^n)^{-1}\), because the middle expression isn’t defined at all until you decide that it can be equal to either \((x^{-1})^n\) or \((x^n)^{-1}\) without causing problems. When proving that a subset \(H\) of a group \(G\) is or is not a subgroup, you may sometimes find it easier to use the conditions \(x,y\in H\implies xy\in H\) and \(x\in H\implies x^{-1}\in H\) instead of the slicker \(x,y\in H\implies xy^{-1}\) in \(H\). When your group \(G\) is abelian, it is usual to write \(x+y\) instead of \(xy\) and \(-x\) instead of \(x^{-1}\). This avoids having to write things like \(2^{-1}=-2\) when working with the integers under addition.
Week 9
A group is an example of a set with additional structure; other examples include fields and rings, and you will see many more. The formal definitions of “subgroup” and “homomorphism” are intended to capture those subsets of groups and functions between groups that understand and respect the additional structure—in this case the binary operation. A subgroup becomes a group in its own right with respect to the same binary operation, and a homomorphism “commutes with the operations” in the following sense. Let \(\varphi\colon G\to H\) be a function, and \(g_1,g_2\in G\). You can either multiply \(g_1\) and \(g_2\) in \(G\), and then apply \(\varphi\), to get \(\varphi(g_1g_2)\), or you can apply \(\varphi\) first and then multiply in \(H\) to get \(\varphi(g_1)\varphi(g_2)\). The map \(\varphi\) is a homomorphism precisely when these two procedures give the same answer, so \(\varphi\) relates the operation on \(G\) to that on \(H\).
Week 10
You now have enough results about groups to start using them to make your proofs quicker (remember that you don’t need to reprove results when you use them). In particular, this week’s problem sheet involves regular usage of the result that every subgroup of a cyclic group is cyclic, and of Lagrange’s theorem. More generally, you use that if \(G\) is a group with subgroup \(H\), then \[|G|=|H|\cdot|\{xH:x\in G\}|,\]so that in particular \(|G|\) is divisible by both \(|H|\) and by the number of distinct cosets of \(H\).
Week 11
This week we mostly talked about the exam. The past papers can be found here—choose “Mathematical Sciences” for the department, and then scroll down to Algebra 1A. Good luck!