In the last week of January and the first several days of this week, I worked through the chapter on subgroups and began the chapter on cyclic groups. I found most of the exercises from the subgroups chapter to be fairly straightforward, except for a couple that we discussed in our last meeting. I’ve included the interesting one below.
Finding the orders of the elements (in this notation, |a| denotes the order of element a) was not a challenge—in fact, it’s really a mechanical process. The tricky part was finding the relationship between the orders, like the last part of the question prompts. The orders are as follows:
- |6|=2, |2|=6, |6+2|=3
- |3|=4, |8|=3, |3+8|=12
- |5|=12, |4|=3, |5+4|=4
There are some patterns that stand out right away, but none are quite precise enough to be the relationship that the book is likely looking for. For example, note that in every set, the orders of two of the elements multiply to give the order of the third. The issue is that there is no pattern to which ones multiply together and which one is given by multiplication of the other two. Another pattern is that all of the orders divide the order of the group. But this isn’t a relationship between the orders of a, b, and a+b, and is also fairly straightforward, and is covered in the next chapter on cyclic groups (which I’m working through right now).
We also looked for a relationship involving some sort of greatest common divisor involving 12, a, and b, but couldn’t find anything there. We ended the meeting thinking that the pattern perhaps was that |a+b| divides |a||b|, in other words, that the order of the sum divides the product of the orders.
I did some more research after the meeting on my own, looking into whether or not, given the orders of a and b, we can find the order of ab (note here I just switched to multiplicative notation, but it’s really the same as a+b was in the integers mod 12, because I’m just omitting the group operation). It turns out that we can’t, in general. we can’t say anything about the order of the products. I can’t understand the proof, because it relies on ring theory and some basic linear algebra (involving groups of matrices), but maybe by the end of the study I will be able to tackle it… Regardless, the result I found is Theorem 1.64 in this document, which I encountered via this MSE post.