Register to reply

Subsets of symmetric groups Sn

by HeronOde
Tags: groups, subsets, symmetric
Share this thread:
HeronOde
#1
Oct30-11, 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.
Phys.Org News Partner Science news on Phys.org
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
spamiam
#2
Oct30-11, 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 "sub-object" for isomorphic to make sense.)
Quote Quote by HeronOde View Post
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: [itex]A := \{\sigma \in S_5 : \sigma(5) = 5 \}[/itex]. Then [itex] S_4 \cong A [/itex] and [itex] A \subseteq S_5 [/itex]. 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
#3
Oct30-11, 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 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".


Register to reply

Related Discussions
Dihedral and Symmetric Groups Calculus & Beyond Homework 5
Center of Symmetric Groups n>= 3 is trivial Calculus & Beyond Homework 3
Fundamental groups of subsets of S^3 Calculus & Beyond Homework 0
Conjugates in symmetric groups Calculus & Beyond Homework 3
Symmetric Groups Calculus & Beyond Homework 5