Assignment Result – Groups

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.

Adrian

…is learning mathematics.