What is the definition of $Alt_\omega$, there is a subgroup of even permutations of finite support, is that what you mean? Why don't you ask it for $S^{fin}_\omega$ first?

Consider the polynomial $g(x)=x(x-1)(x-2)\cdots(x-n)+1$ and the number $N = {\rm lcm}(1,2,\cdots,n)$. Then $N$ is of size about $\exp(n)$ and one can take the remainder of the coefficients of $g$ modulo $N$ to obtain a monic polynomial $f$ with coefficients bounded by $\exp(n)$. This polynomial has no roots in prime fields for all $p\leq n$. Maybe there is a similar construction that takes also care of field extensions.

A $n$-generated group $G$ is free (on those $n$ generators) if and only if $\beta_1^{(2)}(G)=n-1$. In any other case it is $\beta_1^{(2)}(G)<n-1$ and hence $\chi(G)>1-n$ if $G$ is $2$-dimensional since $\chi(G)$ is the alternating sum of $\ell^2$-Betti numbers. It's a proof that is not using the lower central series.

Do you mean $\leq$-preserving or $<$-preserving?

Upper density? en.wikipedia.org/wiki/Natural_density

@IgorRivin: It is good to have a reference, but I was just thinking that Landau looked at the maximal order of elements in Sym(n) in 1903 because this was a difficult and interesting problem (and the solution relies on the prime number theorem). Those people must have thought about the same problem for linear groups, which is much easier.

@IgorRivin; that it true.

It was probably known to Frobenius or anyone who knew how to study characteristic polynomials (and asked the question).