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

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