Why when applying symmetries do you read from right to left? I am new to group theory, and I was just curious why the convention for applying symmetries was to read from right to left? Most mathematical operations I am aware of are read from left to right.
 A: There is a rough convention in mathematics (and there will be lots of exceptions to this!) about when we think of applying group elements left to right and when right to left. Two really important examples of groups that you might have met are matrix groups (ie groups where each group element is a matrix), and permutation groups (groups where you think of each element as a permutation, or shuffling around elements).
An $n\times n$ matrix is a function from $R^n\rightarrow R^n$. Now we typically apply functions from right to left: ie $f\circ g \circ h (x) = f(g(h(x))))$, ie we apply $h$ first, then $g$ then $h$. So if I was looking at a matrix group and I saw $fgh$, I would typically expect to apply $h$ first.
The other way of thinking of groups is as permutations, and these are typically written left to right instead. We sometimes use greek letters for permutations. Let $\pi$ be swap the 1st and 2nd item, and let $\rho$ be swap 2nd and 3rd. Then if I write $\pi\rho$, I'd typically expect this to mean swap 1st and 2nd, and then swap 2nd and 3rd, ie apply $\pi$ first. Eg
ABC <-- start with this
BAC <-- apply $\pi$
BCA <-- apply $\rho$
Which way is more right? Well it depends how we are thinking of a group. And confusingly (or perhaps interestingly), every finite group can be thought of as a matrix group, or as a permutation group, so there is always a way of thinking of it as one way or the other! A tad confusing, but it's nice to have options!
