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My question is this; Prove that the action of $S_n$ on $Y_k$ and $Y_{k} =$ $\dbinom{X}{k}$, where $X=\{1,2,...,n\}$ is faithful for all $1< k < n$.

I know that any action is just fundamentally a permutation therefore we have that all actions are injective, but I am having difficulty trying to express this idea in a rigorous way.

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    I am little about exactly what set $S_n$ is acting on? What is the set? How do you define $T_k$? – 2017-02-19
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    So $\dbinom{X}{k}$ is probably the set of $k$-elements subsets of $X$. But $T_{k}$? Perhaps ordered $k$-tuples? – 2017-02-19
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    The problem here is that you have said "the action", but you have not told us which action. The same group can act on the same set in many different ways. – 2017-02-19
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    sorry, a typo guys. The first question is meant to say Y_k. @AndreasCaranti Y_k is the set of ordered k-tuples such that no numbers are the same. – 2017-02-19

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A good strategy is often just write out exactly what every statement you are given or want to prove means. You want to prove the action of $S_n$ on $Y_k$ is faithful. This means that if $\sigma\in S_n$ is not the identity, then it does not act by the identity: that is, there is some $A\in Y_k$ such that $\sigma(A)\neq A$.

So suppose you have $\sigma\in S_n$ which is not the identity. You want to find some $k$-element subset $A\subset X$ such that $\sigma(A)\neq A$. You're going to need to use the fact that $\sigma$ is not the identity, which mean that there is some $i\in X$ such that $\sigma(i)\neq i$. So maybe we should try using this $i$ to find our set $A$. For instance, suppose we wanted to pick a set $A$ such that $i\in A$. How should we pick the remaining $k-1$ elements of $A$ to be sure that $\sigma(A)$ will not be equal to $A$ (using the fact that $\sigma(i)\neq i$)?

More details are hidden below.

If we can guarantee that $\sigma(i)\not\in A$, then we'll be done, since $\sigma(i)\in\sigma(A)$. So we need to pick our remaining $k-1$ elements so that none of them are equal to $\sigma(i)$. We can do this since $k1$; we just need $k>0$ (so that we are allowed to have $i\in A$ to begin with!).

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    I cannot decide how to pick the remaining k-1 elements. As we are looking at things that are just permutations of the elements of the tuples of Y, do we not just choose k-1 elements to be distinct permutations of all elements in S_n? – 2017-02-19
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    I really can't understand what you're talking about. I think you're confused about some definitions. We are trying to construct an element $A\in Y_k$, which means that $A$ is a subset of $X$ which has $k$ elements. So our $k-1$ remaining elements should be elements of the set $X$. – 2017-02-19
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    Oh I see... I think I was getting at what you were saying but just expressing it very badly. Thank you for all your help. I decided to answer this question by exploring the kernel in the end. – 2017-02-19