Why is symmetric group not matrix? I have just learned from mighty Wikipedia that the symmetric group of an infinite set is not a matrix group. Why?
I can see rep-theory reasons: sizes of minimal non-trivial, non-sign representations of $S_n$ grow as $n$ grows. But I believe that there should really be an elementary linear algebra argument for this. Is there one?
 A: This might not be really elementary, but ... We know that $M_n(k)$ satisfies polynomial identities. For instance, the Amitsur-Levitski Theorem tells us that
$$\sum_{\sigma\in S_{2n}}\epsilon(\sigma)A_{\sigma(1)}\cdots A_{\sigma(2n)}=0_n,
\qquad\forall A_1,\ldots,A_{2n}\in M_n(k).$$
On the contrary, the symmetric group of an infinite set does not satisfy polynomial identities.
A: Edit:  My original idea doesn't work, but unknown's does.  Here are the details.
Let $k$ be a field, which is WLOG algebraically closed.  Let $V$ be a finite-dimensional representation over $k$ of dimension $n$.  Then $S_{\infty}$ contains $(\mathbb{Z}/p\mathbb{Z}))^{p^n}$ (in fact any finite group) as a subgroup, where $p \neq \text{char}(k)$.  Let $g_1, ... g_{p^n}$ be its generators.  Then by "elementary linear algebra" $V$ is a direct sum of $1$-dimensional irreps of $\langle g_1 \rangle$ which $g_2, ... g_{p^n}$ must preserve, while still having order $p$.  But there are only $p^n - 1$ nontrivial ways to do this; hence either one of the $g_i$ acts as the identity or two of them are the same and $V$ cannot be faithful.
Of course, whether this is "elementary linear algebra" or representation theory is debatable, and I think irrelevant.  All I did was find a sequence of finite groups such that the dimension of the smallest faithful representation goes to infinity and I could have done this any number of ways, e.g. I could have chosen $\text{PSL}_2(\mathbb{F}_q)$.
A: The symmetric group of an enumerable set contains every enumerable group as a subgroup. Its linearity would thus imply the linearity of every enumerable group which is false for example 
by the Tits's alternative.
