TITLE: The symmetric group theory of natural numbers
QUESTION [38 upvotes]: Sometimes it is not easy to formulate a correct question. Here is a better version of this  question (I still do not know if it is optimal, but it is better than the previous one).
We say that a set $X$ of natural numbers is symmetric group definable if there exists a first order formula $\theta$ in the group signature such that a symmetric group  $S_n$ satisfies  $\theta$ if and only if $n\in X$. Of course finite sets and sets with finite complements are symmetric group definable. It is not completely trivial to find any infinite set of natural numbers with infinite complement which is symmetric group definable. But it is not a difficult exercise to show that the set of numbers $n$ such that either $n$ or $n-1$ is prime is such a set. 
Question. Is any of the following sets symmetric group definable?
1) the set of even numbers
2) the set of prime numbers
A more vague question Is there a characterization of the Boolean algebra of symmetric group definable sets?
Update Noam D. Elkies' answer below shows that the Boolean algebra contains many sets. Noah Schweber's answer of the previous question suggests looking at the computational complexity of $X$. Clearly, checking whether $n$ belongs to a symmetric group definable set $X$ takes time at most $(n!)^k$ where $k$ is the number of quantifiers in the corresponding formula $\theta$. So we have a couple of even better questions. 
Question 1 Is the converse true, that is every set of numbers recognizable by deterministic a Turing machine in time $\le (n!)^k$ for some $k$ symmetric group definable? Here the size of a number $n$ is $n$ that is we consider the unary representation of natural numbers. 
Question 2 Let us replace $S_n$ by, say, $SL_n(\mathbb{F}_2)$ and similarly define $SL_n(\mathbb{F}_2)$-definable sets of numbers. Will  the Boolean algebra of all $SL_n(\mathbb{F}_2)$-definable sets coincide with the Boolean algebra of all deterministic polynomial time decidable sets of natural numbers?  
Question 3 Can the statements that "Primes are in P" be proved this way?

REPLY [6 votes]: Almost the same first-order sentence that Noam used to define transpositions defines, in $SL_n(\mathbb F_2)$, a unipotent transvection (i.e. with a single $2\times 2$ Jordan block, and the rest $1 \times 1$).
The sentence is "$g$ has order $2$, and its product with any conjugate of itself has order at most $4$".
Indeed, if $g$ has order $2$ then $g$ is unipotent, with all Jordan blocks of size at most $2$. If it has one of size at most $2$ then $g$ times any conjugate of $g$ fixes a codimension two subspace, hence has order $1,2,3$, or $4$. On the other hand, if $g$ has $k$ Jordan blocks, we can view it as a unipotent matrix with a single Jordan block over $\mathbb F_{2^k}$, so using $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix} \begin{pmatrix} 1 & 0 \\ x & 1 \end{pmatrix}=\begin{pmatrix} 1+x & 1 \\ x & 1\end{pmatrix} $$ we can produce any conjugacy class in $SL_2(\mathbb F_{2^k})$, and in particular an element of order $2^k+1>4$.
I suspect from appropriate finite configurations of transvections one can define the disjoint union of the set of nonzero vectors and the set of nonzero linear functions (it is not possible to separate these because of the inverse-transpose automorphism) but I wasn't able to make it work yet.