  Let $G$ be a triangulation of the sphere with vertex set $V$, such that the
faces of the triangulation are properly coloured black and white. Motivated by
applications in the theory of bitrades, Cavenagh and Wanless defined $A_W$ to
be the abelian group generated by the set $V$, with relations $r+c+s=0$ for all
white triangles with vertices $r$, $c$ and $s$. The group $A_B$ can be defined
similarly, using black triangles.
  The paper shows that $A_W$ and $A_B$ are isomorphic, thus establishing the
truth of a well-known conjecture of Cavenagh and Wanless. Connections are made
between the structure of $A_W$ and the theory of asymmetric Laplacians of
finite directed graphs, and weaker results for orientable surfaces of higher
genus are given. The relevance of the group $A_W$ to the understanding of the
embeddings of a partial latin square in an abelian group is also explained.
