# On a limit of permutation groups

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.