Does $S_n$ refer to an object, or a class?

Question

In typical algebra textbooks, the elements of the structures presented are explicitly defined: for example, the n-th Symmetric Group, $S_n$ may be defined as containing precisely the permutations on the set $\{1, \ldots, n\}$, with function composition as its binary operation.

However, this is not a given, and another author may choose to define $S_n$ as containing permutation matrices, or using $n$-tuples, or whatever.

Thus, there appear to be all these $S_n$'s floating around, which is contrary to ordinary phraseology (i.e "Let $S_n$ denote the n-th symmetric group"). What is meant by such a phrase is something along the lines of: "let $S_n$ denote a group isomorphic to anything else one might imaginably call $S_n$, where in the context of where it will be used, the precise elements of the group do not matter."

But, I would say that this is an incorrect use of the word "the", and a more appropriate word would be "a". However, this all rests on what you believe: whether $S_n$ refers to a class of isomorphic groups, or whether it picks out a very specific "canonical" representation of it. So: which of the two is it?

Answer 1

You are absolutely right ,it is a misuse of the word "the"in the sense of the object satisfying some displayed property . You could say for a set A , any set of n objects, that S(A) (or $S_n(A)$ ) is THE set of all bijections of A . That would satisfy me .

Answer 2

I can't say I've ever seen an author define $S_{n}$ as the set of $n \times n$ permutation matrices, although if you have this set of matrices you may call the set $S_{n}$.

To me, $S_{n}$ is the set of bijections on $\{1,2,\ldots,n\}$ equipped with an operation of function composition. Now, if I have the set of bijections of $\{a_{1}, a_{2}, \ldots, a_{n}\}$ I may call this $S_{n}$ because it's clearly canonically isomorphic to the first object. Or if I have the set of permutation matrices as mentioned. But I think the ambiguity comes not from the fuzziness of what $S_{n}$ is, but rather that it is common to refer to two things as being "the same" when we really mean they are isomorphic in a clear fashion.