Subsets of symmetric groups Sn

[HeronOde]

This is not a homework question, just a question that popped into my head over the weekend.

My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set of reals to be a subset of the complex numbers (R2 with multiplication defined in some fashion, right?), where the reals are of the form (a,0) for a a real number, and so we should also be able to think of S4 as a subset of S5.

 

[spamiam]

You always get into trouble when debating whether "is a subset of" is the same thing as "is isomorphic to a subset of." (I guess I should really say "sub-object" for isomorphic to make sense.)

One argument by analogy is that we consider the set of reals to be a subset of the complex numbers (R2 with multiplication defined in some fashion, right?), where the reals are of the form (a,0) for a a real number, and so we should also be able to think of S4 as a subset of S5.

I find this convincing. Take the subset of permutations that fix 5: $A := \{\sigma \in S_5 : \sigma(5) = 5 \}$. Then $S_4 \cong A$ and $A \subseteq S_5$. I always take "isomorphic" to mean "the same" in math. We could always relabel the elements of a group with crazy names, but does that really make it a different group? I don't think so.

 

[Deveno]

strictly speaking, no, because a function with a domain of {1,2,3,4} is obviously different than a function with domain of {1,2,3,4,5}.

but...yes there are (several!) copies of S4 inside S5, just take all permutations that fix n, for some particular element n of {1,2,3,4,5}.

the analogy with the reals and the complex numbers is apt, in fact, the real number a is quite a different thing than the 2-vector (a,0) (one lives on a line in a 1-dimensional world, one lives on a line inside a 2-dimensional world) but the isomorphism a<-->(a,0) is "transparent" the 0 in the second coordinate just "comes along for the ride".

so, even though we are used to saying N ⊆ Z ⊆Q ⊆ R ⊆ C, strictly speaking these are all "different" things, what we mean is something like:

"an isomorphic copy of N lies in the isomorphic copy of Z that lies in an isomorphic copy of Q that lies in the isomorphic copy of R embedded within the complex plane".

however, if two objects are isomorphic as SETS, the only difference is "they have different names", it's just a labelling issue. sets have very little structure, about the only things (properties) we can get our hands on is membership/containment and cardinality (which is why logic works so well for them: in/out corresponds to true/false, and containment corresponds to "implies").

realize, however, that if we add additional structure, we have extra things to check for:

when we say N ⊆ Q, we usually mean that N is (isomorphic to) a commutative sub-semi-ring of the commutative semi-ring of Z, that Z is (isomorphic to) a sub-domain of the integral domain Q and that Q is (isomorphic to) a subfield of R, and that R is (isomorphic to) a subfield of C (boy, that's a mouthful).

isomorphism is an equivalence relation, and much of the process of abstraction involves treating "≅" as "=".

so when we speak of the group "S4", what we usually mean is: "any group isomorphic to S4", for example we might mean S4 acting on the set {1,2,3,4}, or acting on the set {a,b,c,d}, we really don't care about "the details".