An isomorphism between unitals and between related classical groups (Q6636833)

This paper proves the existence of exceptional isomorphisms between some classical groups in characteristic not \(2\), and also isomorphisms between their incidence structures.\N\NLet \((K,\sigma)\) be a division ring with a non-identity anti-automorphism of period two and let \(h:V\times V\rightarrow K\) be a nondegenerate Hermitian form where \(V\) is a left (finite-dimensional) vector space over \(K\). The set of \textit{absolute points} \(U_h\) is defined as the set of all isotropic lines in \(V\) with respect to \(h\). A \textit{secant} is a two-dimensional subspace \(L\) of \(V\) that intersects \(U_h\) in more than two points. If \(L\) is a secant, then the ``trace'' \(U_h\cap L\) is called a \textit{block} of \(U_h\). The set of all blocks is denoted by \(B_h\). In the case where the Witt index of \(h\) is \(1\), the system \((U_h,B_h)\) is called a Hermitian unital with respect to \(h\). As two absolute points belong to a unique block, \((U_h,B_h)\) is, in some sense, an \textit{incidence structure}. If \(h\) is trace-valued, that is \(h(v,v)\) is contained in \(\{\lambda+\sigma(\lambda): \lambda\in K\}\), then the author proves that if an absolute point lies in \(U_h\cap L\), then one has necessarily \(P<L\nleq P^\perp\), and vice-versa. With the same hypotheses (i.e., the same condition on Witt index and trace-valued nature of \(h\)), it is also proved that the projective unitary group \(\mathrm{PU}(V,h)\) acts \(2\)-transitively on \(U_h\) For a quadratic extension \(C/R\) of (commutative) fields with nontrivial Galois group that is generated by \(\sigma\), with some hypotheses, the author constructs a quaternion division algebra \(H\) with center \(R\) whose maximal subfield is \(C\). The author defines two Hermitian unitals, one over \((C,\sigma)\) in a \(4\)-dimensional vector space, and another over \((H,^-)\) where \(^-\) denotes the canonical involution of \(H\), in a \(3\)-dimensional vector space with appropriate Hermitian forms \(g\) and \(h\) of index \(1\). For the case where the characteristic of \(R\) is not two, the existence of an isomorphism between these unitals (isomorphism in the sense of incidence structures) is proved. Finally, an isomorphism between projective groups \(\mathrm{PEU}(C^4,g)\) and \(\mathrm{PEU}(H^3,h)\) associated with groups \(\mathrm{EU}(C^4,g)\) and \(\mathrm{EU}(H^3,h)\) generated by unitary transvections is proved.