# How do group symmetries work exactly?

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.

• Regarding your edit and the various answers, this particular notation that you have reported from your course is very nonstandard, and will be dangerously confusing if you compare to just about everything else in the literature. Commented Feb 27, 2023 at 16:28
• @LeeMosher Huh... that is kinda odd. Even wikipedia shows this particular notation of a permutation and it is not known for being the most inclusive source for math. Commented Feb 27, 2023 at 16:32
• Can you provide a precise citation for wikipedia? Commented Feb 27, 2023 at 16:33
• The one-line notation is listed on Wikipedia at en.wikipedia.org/wiki/Permutation#One-line_notation. It is definitely less common than the other two, especially in group theory, but it is certainly used and reasonable in some contexts, for example in pattern avoidance (en.wikipedia.org/wiki/Permutation_pattern). Anyway bottom line is be aware of what your notation means. Commented Feb 27, 2023 at 21:25

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 $$(4\ \ 1\ \ 2\ \ 3)$$.
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.
$$(a b ... z)$$ 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 $$a,b,c,d$$.