Remark on a theorem of L. Dickson
It is shown that the methods used by \textit{L. Dickson} for a description of the subgroups of the group \(PSL_ 2(q)\) [see: Linear groups (1901)] and Schur's theorem permit to describe periodic linear groups of degree 2 over an arbitrary field. The main theorem implies, in particular, the following assertion. Let A, B be non-zero additive subgroups of a field K of characteristic \(p>0\), and the smallest subfield F containing the product AB is periodic. Then the group \(G=gp(\left( \begin{matrix} 1\quad A\\ 0\quad 1\end{matrix} \right),\left( \begin{matrix} 1\quad 0\\ B\quad 1\end{matrix} \right))\) is either conjugate to \(SL_ 2(F)\) by a diagonal element of \(GL_ 2(K)\) or \(| A| =| B| =p=2\) and G is dihedral, or \(AB=iGF(3), i^ 2=-1\), and the image of G in \(PSL_ 2(K)\) is isomorphic to the alternating group \(A_ 5\); in all cases, G contains an element of the form \(\left( \begin{matrix} 0\quad t\\ -t^{-1} 0\end{matrix} \right)\).
scientific article; zbMATH DE number 3861435