On pseudo-real finite subgroups of \(\mathrm{PGL}_3(\mathbb{C})\)

Let \(G\) be a finite subgroup of \( \rm PGL_3(\mathbb C)\), and let \(\sigma\) be the generator of Gal\((\mathbb C/ \mathbb R)\). We say that \(G\) has a real field of moduli if \(\sigma G\) and \(G\) are \( \rm PGL_3(\mathbb C)\)-conjugates. Furthermore, we say that \(\mathbb R\) is a field of definition for \(G\) or that \(G\) is definable over \(\mathbb R\) if \(G\) is \(\textrm{PGL}_3(\mathbb C)\)-conjugate to some \(\acute{G} \,\subset \, PGL_3(\mathbb R)\). In this situation, we call \(\acute {G}\) a model for \(G\) over \(\mathbb R\). On the other hand, if \(G\) has a real field of moduli but is not definable over \(\mathbb R\), then we call \(G\) pseudo-real.

In this paper, we first show that any finite cyclic subgroup \(G = \mathbb Z / n \mathbb Z\) in \( \rm PGL_3(\mathbb C)\) has a real field of moduli and we provide a necessary and sufficient condition for \(G = \mathbb Z / n \mathbb Z\) to be definable over \(\mathbb R\); see Theorems 2.1, 2.2, and 2.3. We also prove that any dihedral group \(D_2n\) with \(n \geq 3\) in \( \rm PGL_3(\mathbb C)\) is definable over \(\mathbb R\); see Theorem 2.4. Furthermore, we study all other classes of finite subgroups of \( \rm PGL_3(\mathbb C)\), and show that all of them except \(A_4n\), \(A_5n\) and \(S_4n\) are pseudo-real; see Theorems 2.5 and 2.6. Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry; see Theorem 2.7 and Example 2.8. As a result, we can say that if \(G\) is definable over \(\mathbb R\), then its Jordan constant \(J(G)\) = 1, 2, 3, 6 or 60.