Let $E subseteq {1,2,dots}$ be a set of natural numbers such that for any $p,q in E$ there exists $r in E$ satisfying $p | r$ and $q | r$ (where $|$ is the ”divides” relation). Let $S_p$ be the set of all bijective functions in ${0,dots,p-1}$; the symmetric group on $p$ elements.

For $p,q in E$ with $p|q$, we define the injective morphism $phi_{p,q}colon S_p to S_q$ as follows: if $sigmain S_p$, $j in {0,dots,q-1}$, then we write $j = kp+ell$, with $k geq 0$, $ell in {0,1,dots,p-1}$, and put

$$ phi_{p,q}(sigma)(j) = sigma(ell)+kp. $$

Equivalently, $phi_{p,q}(sigma)$ is the unique element $tau in S_q$ such that $tau(j) = sigma(j)$ for all $j in {0,dots,p-1}$ and $tau(j+p) = tau(j)+p$ for all $j in {0,dots,q-1}$. Clearly, $phi_{p,q}(sigma) in S_q$ and $phi_{p,q}$ is an injective morphism.

Observe that by the condition of the first paragraph, ${phi_{p,q} : p,qin E, p | q}$ is an direct system, and so the direct limit

$$mathcal{S} :!= varinjlim_{pin E}{S_p} $$

exists.

My questions: **is this a ”known” group? is there a neat description of this group?**

When $E$ is finite, then $mathcal{S}$ is simply $S_{max(E)}$, so we can suppose that $E$ is infinite.