In a projective plane $PG(2,\mathbb K)$ defined over an algebraically closed field $\mathbb K$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614-1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672-688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa's 3-nets (Adv. Geom. 10:287-310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If $p$ is larger than the order of the group, the above classification holds in characteristic $p>0$ apart from three possible exceptions $\mathrm{Alt}_4$, $\mathrm{Sym}_4$, and $\mathrm{Alt}_5$. Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069-1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469-488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672-688, 2008; Yuzvinsky in Compos. Math. 140:1614-1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641-1648, 2009).

51E20

3-net, dual 3-net, projective plane, embedding, cubic curve