Question

Let \(k\) be a positive integer with \(k \leq |S_n|\). The symmetric group \(S_n\) acts on the set B consisting of all k-element subsets of \({1, \ldots , n}\) by \(\sigma \cdot {a_1, \ldots, a_k}={\sigma(a_1),\ldots,\sigma(a_k)}\).

Determine for which values of \(k\) the actions of \(S_n\) on k-element subsets is faithul

Answer 1

Let k be a positive integer with \(k \leq n\). The symmetric group \(S_n\) acts on the set \(B\) consisting of all \(k\)-element subsets of \(\{1,…,n\}\) by \(\sigma⋅\{a_1,\ldots,a_k\}=\{σ(a1),\ldots,σ(ak)\}\).


Determine for which values of \(k\) the actions of \(S_n\) on \(k\)-element subsets is faithul

Answer 2

Egh, I can't type: \(\{\sigma(a_1), \ldots , \sigma(a_k)\}\)

Answer 3

\(S_n\) is a group and it's operation is symmetric?

Answer 4

\(S_n\) is the symmetric group of the set \(\{1, \ldots, n\}\)

Answer 5

I know that if k = n, the operation is not faithful since any permutation will still give the n-element subset, and that if k = 1 it i's trivially faithful, but I'm stuck on proving whether or not it's faithful for 1 < k < n

Answer 6

How would it not be faithful?

Answer 7

An action is faithful if different elements of \(S_n\) induce different permutations of B, and if k = n, there is only one n-element subset, which every element of S_n maps to.

ie if for different \(g, h \in S_n\), \(g\cdot x\neq h\cdot x\) for some \(x \in B\)

Answer 8

Hmmm, one thing I can say is that the set of all subsets is the power set and it has cardinality \(2^k\)

Answer 9

So is \(S_n\) full of \(n\) mappings which will permute the elements in \(B\)?

Answer 10

\(S_n\) is the set of n! distinct permutations of the set \(\{1, 2, ..., n\}\) it acts on B by permuting it's elements ie:
if we have the permutation (1 2 3) (1 gets send to 2, 2 to 3, 3 to 1) act on {1, 2} we get
\((1\,\,2\,\,\,3) \cdot \{1,2\}=\{2, 3\}\)

Answer 11

Should it result in \(\{3,1\}\)?

Answer 12

the action is defined as \(\sigma \cdot \{a_1, ... , a_k\} = \{\sigma(a_1), ... ,\sigma(a_k)\}\)
so in the case that \(\sigma =\)(1 2 3), \(\sigma \cdot \{1, 2\}=\{\sigma(1), \sigma(2)\}=\{2, 3\}\)

Answer 13

Okay, I think I get that much now.

Answer 14

I've seen this before, but the notation we use for permutation was maybe backwards.

Answer 15

The number of \(k\) element subsets in \(n\) is \(\binom nk\)

Answer 16

It's faithful if no two permutations produce equivalent sets?

Answer 17

Consider what happens with some concrete examples or concrete permutations.

Answer 18

now if k =1, there exists two permutations \(\sigma\) and \(\tau\) such that \(\sigma(a) \neq \tau(a)\) if we let \(x = \{a\}\) (which is a subset of {1, ..., n} of cardinality 1)
then without loss of generality we have \(\sigma \cdot x \neq \tau \cdot x\), so action is trivial if k is one,

if k = n, then there is only one subset of {1, ..., n} namely that set itself so every permutation will give you that subset.

now I think for 1 < k < n I will have to show that the permutation representation is injective.
I think

if we let \(\sigma, \tau\) be different permutations again, then without loss of generality \(\sigma(a_1) = x, \tau(a_2) = x\) then we can choose a subset z of cardinality k so that x and \(a_1\) are in the subset, but \(a_2\) isn't (since k < n, we have at least 1 element not in the subset), then \(\sigma \cdot z \neq \tau \cdot z\), since \(\sigma \cdot z\) contains x, and \(\tau \cdot z\) doesn't, so the action is faithful if k<n

Answer 19

I think this works :D

Answer 20

There only needs to be two distinct permutations which don't result in the same subset?

Answer 21

No, for any two distinct permutations there must be a subset so that \(\sigma \cdot x \neq \tau \cdot x\).
In this proof i think i managed to show that we can always construct at least 1 subset of length k which does this

Answer 22

So if I get this right:
If the permutations are distinct, there has to be an output where the permutations have a different input to output it. You then find a subset which has the input for one of the permutations but lacks the needed input for the second permutation. Thus there is an output which one has and the other doesn't.

Answer 23

yes

Answer 24

and such a subset always exists, since there is at least 1 element from {1, ..., n} not in the subset

Answer 25

You can ensure an excluded input is excluded, and you can also ensure the necessarily included input gets included?

Answer 26

Yes, since we only consider 2 elements, a1 and x, and for 1 < k < n, k is at least 2 :)

Answer 27

What if a1 and a2 are greater than k, does that matter?

Answer 28

Not really, since k is just how much elements are in the subset

Answer 29

ie if we have k = 2, and two permutations \(\sigma(7) = 8\) and \(\tau(8)=8\)
then we choose a subset which doesn't have 7 in it, but contains 8 and any other element, say 5 so we get {5, 8}
now even if the 2 permutations map 5 to the same element, 8 still gets mapped to a different one

Answer 30

two permutations such that*

Answer 31

Thanks for the help by the way, trying to explain it to someone else always helps it seems :)