# Subsets of symmetric groups Sn

• HeronOde
In summary, the analogy between the real numbers and the complex numbers is apt, and S4 is a subset of S5.
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.

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.)
HeronOde said:
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.

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".

## What are subsets of symmetric groups Sn?

Subsets of symmetric groups Sn are collections of elements from the symmetric group Sn that satisfy certain properties. These subsets can include all the elements of Sn, or only a portion of them.

## What are the properties of subsets of symmetric groups Sn?

The properties of subsets of symmetric groups Sn vary depending on the specific subset. Some common properties include being closed under certain operations, such as multiplication or inversion, and containing a specific number of elements.

## How are subsets of symmetric groups Sn used in mathematics?

Subsets of symmetric groups Sn are used in a variety of mathematical fields, including group theory, combinatorics, and algebraic geometry. They can be used to study the symmetries of objects, solve equations, and classify mathematical structures.

## What are some examples of subsets of symmetric groups Sn?

Some examples of subsets of symmetric groups Sn include the set of all even permutations in Sn, the set of all cyclic permutations in Sn, and the set of all permutations with a specific cycle structure. Other examples include subsets defined by certain properties, such as being a subgroup or a coset.

## How do subsets of symmetric groups Sn relate to other mathematical concepts?

Subsets of symmetric groups Sn are closely related to other mathematical concepts, such as groups, permutations, and symmetric polynomials. They can also be used to study other mathematical structures, such as graphs, codes, and designs.

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