  In this paper we investigate properties of algebraic varieties representing
group-based phylogenetic models. We propose a method of generating many
phylogenetic invariants. We prove that we obtain all invariants for any tree
for the binary Jukes-Cantor model. We conjecture that our method can give all
phylogenetic invariants for any tree. We show that for 3-Kimura our conjecture
is equivalent to the conjecture of Sturmfels and Sullivant. This, combined with
the results of Sturmfels and Sullivant, would make it possible to determine all
phylogenetic invariants for any tree for 3-Kimura model, and also other
phylogenetic models. Next we give the (first) example of a non-normal general
group-based model for an abelian group. Following Kubjas we also determine some
invariants of group-based models showing that the associated varieties do not
have to be deformation equivalent.
