How do group symmetries work exactly?

Question

Currently in my discrete mathematics course at the university I study at, we are on the topic of permutations and groups. We have learned that a group $G$ can be expressed in terms of a set of generators $S$ like this $G=\langle S\rangle$ such that all elements can be expressed using elements of $S$ and the operation under which $G$ is closed. However, in some examples in our lecture notes and in certain assignments, geometric groups containing a square with vertices $1,2,3,4$ are defined using $\langle (2\text{ }4),(1\text{ }2 \text{ }3 \text{ }4)\rangle$. I understand that $(2\text{ }4)$ is used to signify reflections, but why the identity permutation is used for signifying rotations is a mystery to me. According to the book, every $90^\circ$ rotation could then be written as $(1\text{ }2\text{ }3\text{ }4)^n$ which seems wrong considering how the identity permutation works. I have looked on the internet for answers but for some reason wikipedia has the exact same definition of a square by its symmetries and I couldn't find anything remotely relevant elsewhere (aside from some article to which I had no access). Is there something I am missing? Wouldn't it make more sense to use the permutation $(4 \text{ }1\text{ }2\text{ }3)$ instead as that alludes to an actual rotation?

Edit: first of all, thanks for all the answers so quickly! Secondly, for this course we learned that the single-line notation of permutations was always the same as the double-lined matrix notation except for the first line just being absent such that $$\left(\begin{matrix} 1 & 2 & 3 & 4 \\ \sigma(1) & \sigma(2) & \sigma(3) & \sigma(4)\end{matrix}\right)= (\sigma(1)\text{ }\sigma(2)\text{ }\sigma(3)\text{ }\sigma(4))$$ which probably led to all the confusion as all other examples in lecture notes either used the above added identity, the disjoint cycle notation or the transposition notation.

Second edit: after the question brought up by @LeeMosher about the non-standard notation, I took another look into the notation section in our lecture notes and apparently I confused normal brackets with square brackets in the one-lined notation (where one-lined notation is $[\sigma(1)...\sigma(n)]$). My apologies for this brain fart.

Answer 1

The notation $(1\ \ 2\ \ 3\ \ 4)$ does not denote the identity transformation. It denotes the cyclic permutation which maps $1$ into $2$, $2$ into $3$, $3$ into $4$, and $4$ into $1$. Which, by the way, is equal to $(4\ \ 1\ \ 2\ \ 3)$.

Answer 2

The two permutation $(1 2 3 4)$ and $(4 1 2 3)$ are the same. In fact in the first you have $1\mapsto 2$, $2\mapsto 3$,$3\mapsto 4$,$4\mapsto 1$ and this coincides with the second permutation.

There is not a unique way to represent a permutation.

Answer 3

$(a b ... z)$ means $a$ goes to $b$, $b$ goes to $c$ ... $y$ goes to $z$ and $z$ goes to $a$. Therefore $(1 2 3 4)$ means exactly the same as $(4 1 2 3)$.

Sometimes $1$ is also used to mean the identity of a group. Here it's being used to label the first of the vertices as well or instead. So a bit confusing! If it makes it easier, replace the vertices $1,2,3,4$ in your mind by $a,b,c,d$.

Hope this helps!