Is P a group under the operation composition of classes?  First, we must define composition of classes.  Taking an element from each class, and joining them by unattaching a strand from each element and joining the ends composes the elements. Clearly composing two classes [p1] and [p2] in P yields a class of paths which is in S and travels from b to b.  (see figure 3)

Also, let [i] be the class of paths which never leaves b.  Observe that composing [i] with any class [p] in P gives the path [p]. (see figure 4)

In addition, note that composition of classes of paths is associative.

Also, for any [p] in P, an element [p]' of P exists for which [p]*[p]' = [i].  In particular, [p]` = [p`], where p` is the inverse of p in P.  That is, p*p`=i.  We know p` exists, because p` is simply the path p traced in the opposite direction.  Notice that for any q in [p], q can be rewritten as p, and hence we can find an element of [p]` that is the inverse of q, namely p`.  Therefore, [p]`=[p`].

So we have shown that P is indeed a group under the operation composition of classes of paths.

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