Subgroups of Sn containing Sn-1

[potmobius]

I want to show that the only subgroups of Sn (the symmetric group of n elements) containing Sn-1 are Sn and Sn-1. So essentially, all that's needed to be checked is that there is no subgroup of order greater than (n-1)!, the order of Sn-1 and less than n!, the order of Sn. I was first thinking that if there did exist such a subgroup of Sn, then it would contradict Lagrange's theorem, as then (n-1)! would not divide the order of said subgroup. However, that fails for large enough n, as the said subgroup may have order (n-1)*k, where 2<=k<n. Any ideas? Or should I just try a different approach? Can induction work here?

 

[micromass]

I don't think Lagrange's theorem is going to work here.

Try to prove that if you have S_{n-1} and if you add an arbitrary other element of S_n, then that generates entire S_n.
So, for example, what you can show is how to form all other elements of S_n using just S_{n-1} and the arbitrary other element.

Things you might want to look at is theorems that say things like "this collection of elements generates S_n". Do you know such things?

 

[potmobius]

I don't know of such theorems. Are you perchance hinting to the set of generators or the set of orbits of Sn?

But I'm thinking that if you take an element in Sn, call it sigma_n that's not in Sn-1, then it must shift the position of the nth element in the set (1,...,n), since if it were in Sn-1, then it could permute elements in the set (1,...,n-1), but would have to leave n in its place. Now, sigma_n can do whatever it pleases to the elements of (1,...,n-1), and I get that the composition of sigma_n with elements of Sn-1 would give you some elements of Sn, but what guarantees that we would get ALL elements of Sn?

 

[micromass]

I don't know of such theorems. Are you perchance hinting to the set of generators or the set of orbits of Sn?

But I'm thinking that if you take an element in Sn, call it sigma_n that's not in Sn-1, then it must shift the position of the nth element in the set (1,...,n), since if it were in Sn-1, then it could permute elements in the set (1,...,n-1), but would have to leave n in its place. Now, sigma_n can do whatever it pleases to the elements of (1,...,n-1), and I get that the composition of sigma_n with elements of Sn-1 would give you some elements of Sn, but what guarantees that we would get ALL elements of Sn?

Ah, that is a nice approach. So indeed, an element f of S_{n-1} has the property that f(n)=n.
Now, we are given a specific element g such that g(n)\neq n. We want to see if we can write any element in S_n using S_{n-1} and g.

Thus: take h\in S_n arbitrary. Can you find f,f^\prime\in S_{n-1} such that h=fgf^\prime (I claim we can always write h in that form). Think about what must happen if you apply both h and fgf' on the element n and on 1,...,n-1.

 

[potmobius]

I don't know how to find such f,f', but my first guess was to define f inverse as f' and f as h restricted to the set (1,...,n-1), but I don't know if that actually works. But the equation you gave reminds me of conjugates. Are you trying to prove that h and g are conjugates? Or am I completely off? Also, I found a theorem saying that any element of Sn can be written as a product of transpositions. Does that help here?

 

[micromass]

I don't know how to find such f,f', but my first guess was to define f inverse as f' and f as h restricted to the set (1,...,n-1), but I don't know if that actually works. But the equation you gave reminds me of conjugates. Are you trying to prove that h and g are conjugates? Or am I completely off?

I wouldn't think about conjugates. Just try to think about what f,g and f' should be. For example, apply both h and fgf' to n. What can you conclude?

Also, I found a theorem saying that any element of Sn can be written as a product of transpositions. Does that help here?

Perhaps. We'll see.

 

[potmobius]

Ok. Either h(n) = n or not. If h(n) = n, then h is in Sn-1, in which case I don't know what g should be, as g(n) not equal to n, and regardless of what f and f' are, fgf'(n) won't equal n, which is a problem (but maybe you just disregard g completely in this case).
Suppose h(n) not equal to n. Then g(k) = k and f(k) = h(k), for 1<=k<=n-1. And f' can just be the identity. I think in that case, h = fgf'. I see no problem with this approach, besides the fact that g is chosen very specifically. Is that allowed? If not, could I just define f(k) = h(g inverse (k)) for all 1<=k<=n-1. If so, why the need for f' at all?

 

[micromass]

Ok. Either h(n) = n or not. If h(n) = n, then h is in Sn-1, in which case I don't know what g should be, as g(n) not equal to n, and regardless of what f and f' are, fgf'(n) won't equal n, which is a problem (but maybe you just disregard g completely in this case).

The goal is to write every element in S_n as a product of elements in S_{n-1} and the specific element g. But if h is in S_{n-1} then this can be done trivially. So in this case, we don't need to look at h=fgf'.

Suppose h(n) not equal to n. Then g(k) = k and f(k) = h(k), for 1<=k<=n-1. And f' can just be the identity. I think in that case, h = fgf'. I see no problem with this approach, besides the fact that g is chosen very specifically. Is that allowed? If not, could I just define f(k) = h(g inverse (k)) for all 1<=k<=n-1. If so, why the need for f' at all?

But g is a fixed element that you add to S_{n-1}. The elements h and g are fixed, and you need to find f and f'.

 

[potmobius]

I still don't see why both f anf f' are needed. But doesn't defining f(k) = h°g^-1(k) for all 1<=k<=n-1 work? If f' is really necessary, it can be the identity, so h = f°g°f'

 

[micromass]

I still don't see why both f anf f' are needed. But doesn't defining f(k) = h°g^-1(k) for all 1<=k<=n-1 work? If f' is really necessary, it can be the identity, so h = f°g°f'

Is hg^{-1} in S_{n-1}?

 

[potmobius]

As it is restricted (can only permute the first n-1 elements), I think it is in Sn-1. But I think that since f is in Sn-1, f(n) = n. But g(n) ≠ n, so (f°g)(n) = n, which is not what we want, as h(n) ≠ n. But I think that one of f and f' should be a transposition (perhaps of order 2?) and the other should be some sort of a rotation (for the lack of a better term) (perhaps of order n?). Am I inching closer to the right track, or going away from it?

 

[micromass]

You're getting closer, I think.

 

[potmobius]

I now see the importance of f. Since g(n)≠n, g(n) = p for some 1≤p≤n-1. Similarly, h(n) = q for some 1≤q≤n-1. Since we have freedom to choose f, we can define it such that f(p) = q. Now, if f' is the identity (I still don't see its importance), then h(n) = (f°g°f')(n). Similarly, we can define (f°g)(t) = h(t) for all 1≤t≤n-1. Is there a flaw in this?

 

[micromass]

I now see the importance of f. Since g(n)≠n, g(n) = p for some 1≤p≤n-1. Similarly, h(n) = q for some 1≤q≤n-1. Since we have freedom to choose f, we can define it such that f(p) = q. Now, if f' is the identity (I still don't see its importance), then h(n) = (f°g°f')(n). Similarly, we can define (f°g)(t) = h(t) for all 1≤t≤n-1. Is there a flaw in this?

Does this also work for the t such that g(t)=n??

 

[potmobius]

I did consider this situation, but didn't at first think it would be problematic. But clearly, h(t) doesn't have to equal n. and now f' would be useful. But I'm still not sure how to do so. Any hints on how to proceed?