Free groups and roses

Consider the 2n symbols x1, . . . , xn, x−1 1 , . . . , x −1 n . Collect these into a set G. There is a map from G to itself which interchanges xi and x−1 i . This map is denoted by the usual inversion symbol g 7→ g−1. Thus x−1 i is unambiguous and (x −1 i ) −1 = xi. The empty word will be denoted by 1. A word is defined to be a finite ordered list of elements of G, depicted as exemplified by x1x 2 x1x1x −1 1 . Two words are said to be equivalent if they are related by a finite sequence of operations wgg−1v 7→ wv or the reverse. Here w and v represent words in and g ∈ G. Thus the word x1x 2 x1x1x −1 1 is equivalent to x1x −1 2 x1. The free group on n generators, Fn, is defined as a set to be the set of equivalence classes of words. The group operation is defined by v ·w = vw. That is, placing the two words side-by-side. The inverse operation is defined by (g1 · · · gk) = g−1 k · · · g −1 1 .