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Meepi
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
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
Egh, I can't type: \(\{\sigma(a_1), \ldots , \sigma(a_k)\}\)
\(S_n\) is a group and it's operation is symmetric?
\(S_n\) is the symmetric group of the set \(\{1, \ldots, n\}\)
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
How would it not be faithful?
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 \(x \in B\)
Hmmm, one thing I can say is that the set of all subsets is the power set and it has cardinality \(2^k\)
So is \(S_n\) full of \(n\) mappings which will permute the elements in \(B\)?
\(S_n\) 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 \((1\,\,2\,\,\,3) \cdot \{1,2\}=\{2, 3\}\)
Should it result in \(\{3,1\}\)?
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), \(\sigma \cdot \{1, 2\}=\{\sigma(1), \sigma(2)\}=\{2, 3\}\)
Okay, I think I get that much now.
I've seen this before, but the notation we use for permutation was maybe backwards.
The number of \(k\) element subsets in \(n\) is \(\binom nk\)
It's faithful if no two permutations produce equivalent sets?
Consider what happens with some concrete examples or concrete permutations.
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
There only needs to be two distinct permutations which don't result in the same subset?
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
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.
and such a subset always exists, since there is at least 1 element from {1, ..., n} not in the subset
You can ensure an excluded input is excluded, and you can also ensure the necessarily included input gets included?
Yes, since we only consider 2 elements, a1 and x, and for 1 < k < n, k is at least 2 :)
What if a1 and a2 are greater than k, does that matter?
Not really, since k is just how much elements are in the subset
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
two permutations such that*
Thanks for the help by the way, trying to explain it to someone else always helps it seems :)