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Subsets of symmetric groups Sn 
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#1
Oct3011, 03:46 AM

P: 1

This is not a hw 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. 


#2
Oct3011, 11:06 AM

P: 366

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 "subobject" for isomorphic to make sense.)



#3
Oct3011, 11:55 AM

Sci Advisor
P: 906

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 2vector (a,0) (one lives on a line in a 1dimensional world, one lives on a line inside a 2dimensional 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 subsemiring of the commutative semiring of Z, that Z is (isomorphic to) a subdomain 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". 


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