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Let $\mathcal{X}$ be a (projective, geometrically irreducible, nonsingular)algebraic curve of genus $g \ge 2$ defined over an algebraically closed field$\mathbb{K}$ of odd characteristic $p\ge 0$, and let $\rm{Aut}(\mathcal{X})$ bethe group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$element-wise. For any a subgroup $G$ of $\rm{Aut}(\mathcal{X})$ whose order isa power of an odd prime $d$ other than $p$, the bound proven by Zomorrodian forRiemann surfaces is $|G|\leq 9(g-1)$ where the extremal case can only beobtained for $d=3$. We prove Zomorrodian's result for any $\mathbb{K}$. Theessential part of our paper is devoted to extremal $3$-Zomorrodian curves$\mathcal{X}$. Two cases are distinguished according as the quotient curve$\mathcal{X}/Z$ for a central subgroup $Z$ of $\rm{Aut}(\mathcal{X})$ of order$3$ is either elliptic, or not. For elliptic type extremal $3$-Zomorrodiancurves $\mathcal{X}$, we completely determine the two possibilities for theabstract structure of $G$ using deeper results on finite $3$-groups. We alsoshow infinite families of extremal $3$-Zomorrodian curves for both types,elliptic or non-elliptic. Our paper does not adapt methods from the theory ofRiemann surfaces, nevertheless it sheds a new light on the connection betweenRiemann surfaces and their automorphism groups.

Large odd prime power order automorphism groups of algebraic curves in any characteristic