"Permutation groups" confused me for the longest time, until someone clearly spelled out to me that the permutation group $S_n$ was the set of all bijections from a set of n elements onto itself.
But that's not what my old interpretation was like, but it was still a group. I viewed the "permutations" in cycle notation as "index swappers". For example, if we are working with the permutations of $12345$:
Cycle operation (145) in my old interpretation was that the element at index 1 goes to index 4, the element at index 4 goes to index 5, and the element at index 5 goes to index 1.
So if the permutation was 34125, then by my old interpretation, applying (145) to it, we would get 54132.
But by the bijection definition:
(145) applied to 34125 sends the elements 1 to 4, 4 to 5, and 5 to 1, which is: 35421.
My older interpretation is still a group, and it's isomorphic to the permutation group (I think), but I don't think it's the permutation group.
Is my recent understanding valid?
Best Answer
There are quite a few technicalities to note. But I'd like to first say that this is a great question since you are challenging your initial understanding of a mathematical concept. It is often very easy to assume we understand a concept without actually critically thinking about the details; so you are doing great.
Let $X$ be a set (of any cardinality). The symmetric group on $X$, denoted $\mathrm{Sym}(X)$, is the group of all bijections on $X$. The symmetric group of degree $n$ (or sometimes called the symmetric group on $n$ letters) is the group $S_n := \mathrm{Sym}(X)$ where $X := \{1,2,\dots, n\}$. This notation is not completely standardized, although quite common. See here for some discussion on $S_n$. Importatntly, if $X,Y$ are two sets of the same cardinality, then $\mathrm{Sym}(X)$ is group isomorphic to $\mathrm{Sym}(Y)$.
As @Cpc clarified, when people say "permutation groups" they are generally referring to a group that is a subgroup of $S_n$. Again, $S_n$ is called the symmetric group of degree $n$.
I'd like to clarify that the elements of $S_n$ are bijections, meaning functions. It is bit difficult to come up with notation to refer to all such bijections in $S_n$. Using $S_5$ for example, one common way is to write $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ a & b & c & d & e \end{pmatrix} $$ to denote the bijection on $\{1,2,3,4,5\}$ where $1 \mapsto a$, $2 \mapsto b$, and so on (where of course $a,b,c,d,e$ are distinct integers from $1$ to $5$). This notation allows you to describe each element of $S_n$ in a unique way.
Let say we are still working in $S_5$. The notation $(1\ 4\ 5)$ is cycle notation, which refers to the particular bijection in $S_5$ where $1 \mapsto 4 \mapsto 5 \mapsto 1$ and leaves all other numbers (namely $2,3$) untouched. There are a few important things to note. The cycles $(1\ 4\ 5)$, $(4\ 5\ 1)$, and $(5\ 1\ 4)$ all denote the same bijection in $S_5$. More generally, you can cyclically permute the numbers in the cycle notation and still refer to the same bijection. Moreover, and very importantly, not all elements of $S_5$ (or any $S_n$ for $n > 3$) are cycles. Consider the permutation $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 3 & 5 & 4 \end{pmatrix} $$ which cannot be written as a cycle (try it). Yet, it can be written as the composition of cycles: $(1,2) \circ (5,4)$ meaning first transpose $5,4$ and then transpose $1,2$. People typically write $(1,2)(5,4)$ instead of $(1,2) \circ (5,4)$. Now, although not all elements of $S_n$ are cycles, it is a theorem that all elements of $S_n$ may be written as the composition of cycles which deal with distinct elements. Again, see here (in particular the section labelled "cycles"), to learn more about cycle notation.
The reason why I have clarified the notation is that you are seeming to confuse elements of $S_5$ which are functions with perhaps ordered $5$-tuples of the numbers $1, \ldots, 5$. When you write "if the permutation was $34125$, then by my old interpretation, applying $(145)$ to it, we would get $54132$", it seems as though you are viewing the elements of $S_5$ to be the ordered tuples $(3,4,1,2,5)$ and $(5,4,1,3,2)$ while your cycle notation of $(145)$ is a function on the "elements" of $S_5$ rather than an element of $S_5$ itself. At least that is how I am interpreting your post. So again, I'd like to emphasize that $S_5$ is the group of bijections (functions) on the set $\{1,2,3,4,5\}$ and not the set of ordered $5$-tuples. Maybe I'm interpreting your post wrong though; maybe when you write "$34125$" you mean the function on $\{1,2,3,4,5\}$ which sends $1 \mapsto 3$, $2 \mapsto 4$, $3 \mapsto 1$, $4 \mapsto 2$, $5 \mapsto 5$ (in which case this is a condensed version of the notation I presented in point 3 above). Either way, I hope the prior points help clear up the notation that is commonly used and what the elements of $S_5$ are defined to be.
Not sure if this is helpful, but maybe the following is a group that you are getting confused with $S_5$. Consider $\mathrm{Sym}(X)$ where $$ X := \left\{ (a,b,c,d,e) : a,b,c,d,e \text{ are distinct integers from } 1 \text{ to } 5 \right\}. $$ Then your example of $(145)$ is a way of denoting the element in $\mathrm{Sym}(X)$ which is the bijection where $$ (a,b,c,d,e) \mapsto (e,b,c,a,d). $$ Note that $\mathrm{Sym}(X)$ is not group isomorphic to $S_5$ since $|S_5| = 5!$ while $|\mathrm{Sym}(X)| = (5!)!$.
I hope this helps clear stuff up! It seems that the main issue are just dealing with formal definitions and notation. Everyone please feel free to help refine my explanations or comment on anything I missed.
edit1: quick fixes following @DerekHolt's comment