Suppose we have the group $ G $ of invertible functions over a particular set $ S subseteq mathbb {R} $ under composition. I am interested in the divisibility of such a group. Take, for example $ S =[-1, 1]$for each $ n in mathbb {N} $you could try to find a function $ f in S $ so that

$$ underbrace {f circ f cdots circ f} _ {n text {times}} = sin ( frac { pi x} {2}) $$

In general we want to satisfy $ forall n forall x exists y y ^ n = x $, the usual axiom of divisibility. I suspect that these groups are not divisible, but it is known that they can always be embedded in a divisible group $ overline {G} $, The construction of $ overline {G} $ I've read that a series of wreath products and a direct limit are required. This usually makes things very abstract and can be difficult to identify $ f $ as an element of $ overline {G} $,

Is there a better way to visualize this construction? My goal is to understand the elements in $ overline {G} $ as features over some larger (possibly not even real) $ S supseteq S $,