Finite phylogenetic complexity of $Z_p$ and invariants for $Z_3$

We study phylogenetic complexity of finite abelian groups - an invariant introduced by Sturmfels and Sullivant. The invariant is hard to compute - so far it was only known for $Z_2$, in which case it equals $2$. We prove that phylogenetic complexity of any group $Z_p$, where $p$ is prime, is finite. We also show, as conjectured by Sturmfels and Sullivant, that the phylogenetic complexity of $Z_3$ equals $3$.