Free products and wild nearfields à la B. H. Neumann and Hans Zassenhaus (Q470243)

A nearfield \((N,+,\cdot)\) is called Dickson if there exists a skew field multiplication ``\(\;\;\)'' on \((N,+)\) and a map \(\alpha:N^{\times}\to \Aut(N,+,\;)\) such that \(a\cdot b=ab^{\alpha(a)}\). Non-Dickson nearfields are called wild.NEWLINENEWLINEThe author recalls a construction of B.~H.~Neumann of subgroups \(H\) of the group \(\mathrm{SL}_2\mathbb Z\) that induce sharply transitive groups on the projective line over the rationals. The product of such a group with the multiplicative group of the field of rationals yields the multiplicative group of a wild nearfield.NEWLINENEWLINEIn a similar spirit and inspired by \textit{H. Zassenhaus} [Result. Math. 11, 317--358 (1987; Zbl 0649.12018)] the author shows the existence of subgroups inside \(\mathrm{PSL}_2E\), which act sharply transitively on the projective line over \(E\). Here \(E\) is a purely transcendental extension of yet another field \(F\). Under suitable conditions this again leads to wild nearfields.