Assignment Result - Groups

December 24, 2021 Adrian

My second assignment on groups was marked and returned. As predicted, there's a lot to take away from this.

Basic Groups

At this level it seems it's not sufficient to use notation like $pH$ , $qH$ and $rH$ to refer to subgroups. In my head, I know what the binary operator of these subgroups is but for the benefit of the reader the (better) convention is to be explicit with the binary operator: $p+H$ , $q+H$ , and $r+H$ specifically.

Definition check: a cyclic group is just a group that contains a generator that generates the whole group. It doesn't have to "cycle" back to the beginning of the ordered set. eg, infinite sets can form a cyclic group under addition.

Mistakes in basic proofs. This one is a classic. With $p,q\in \mathbb{Q}$ , I had to prove $\phi(q)=\frac{3q}{2}$ was onto. My answer:

$\forall\: q\in\mathbb{Q},\: \exists\: p\in\mathbb{Q},\: \text{s.t.}\: \phi(p)=q,\: \therefore \phi$

Which I was all smug about because it looks pretty. But this is just the definition of the function being onto and I'd neglected to state the actual $p$ that always exists. Namely $p=\frac{2q}{3}$ .

Group Classification

Group decompositions can't be written as $\mathbb{Z}_{6} \times \mathbb{Z}_{45}$ . It needs to be written as $\mathbb{Z}_{3} \times \mathbb{Z}_{6} \times \mathbb{Z}_{15}$ . 3 and 15 are not coprime, so cannot be combined.

Finite Groups

As a part of a group presentation, there are kind of identities like $sr=r^{5}s$ . However, for the most part in your answer, especially when you're talking directly about elements being in your group, they must always be in the form $r^{i}s^{j}$ (with $r$ first).

In being asked to deduce an isomorphism, I heavy-handedly defined both groups (as they were small) and then also stated a function explicitly that would define the isomorphism. Apparently it's just enough to state that both groups were of order 2. Therefore they're isomorphic.

I need to become a lot more familiar with the concept of centres of a group. To show that a subgroup is the centre of its parent group, I needed to show that the subgroups elements were centres of the subgroup. I was was just trying to wing it, and although I essentially got the answer right, I showed I had a complete misunderstanding of centres.

Chapter 10 of 24 - Quadratic Reciprocity

December 20, 2021 Adrian

Fun chapter. Squares are very important, turns out. Chapter was about methods of solving quadratic formulas modulo some prime.

Covered Euler's Criterion for quadratic residues, The Legendre symbol, Gauss's Lemma, and then the law of quadratic reciprocity.

I'll be looking at rings next. I've been waiting to investigate these things for a while...

Pure Maths

Chapter 9 of 24 - Multiplicative Functions

December 18, 2021 Adrian

Multiplication! That sounds nice and simple doesn't it! Unfortunately, that's not what that title means.

Any number can be broken down into its prime decomposition ( $15=3\times 5$ ). A multiplicative function is where a function of a number is equal to the function of each prime in the prime decomposition of that number all multiplied together. So if $n=p_1\times p_2$ , where p is a prime in the prime decomposition and if $f(n) = f(p_{1}\times p_{2}) = f(p_{1})\times f(p_{2})$ , then the function $f$ is a multiplicative function.

Next section went into detail about perfect numbers. A perfect number is equal to the sum of its positive factors excluding itself. Best example here is the perfect number 6=1+2+3.

The rest of the chapter was introducing Euler's $\phi$ -function (also multiplicative), and using this new function, the introduction of primitive roots.

Adrian

Monthly Archives: December 2021

Chapter 10 of 24 - Quadratic Reciprocity

Chapter 9 of 24 - Multiplicative Functions

…is learning mathematics.