Finite subgroups of PGL₂(K) and their invariants

This thesis looks at finite subgroups of the projective group of 2 x 2 matrices over a skew field and the invariants of these subgroups. Chapter 0 recalls most of the preliminary results needed in subsequent chapters. In particular the construction of Kk(x) is. outlined briefly. Chapter 1 establishes an isomorphism between the group of tame automorphisms in one variable over the skew field K and the projective group of 2 x 2 matrices over K, PGL2(K). It shows that if K is of suitable characteristic, then any element A of PGL2(K) of finite order has either two or else infinitely many fixed points in some extension of K. In particular this means that such A can be diagonalized. Chapter 2 is divided into three sections. The first section deals with finite subgroups of PGL2(K) whose elements may have infinitely many fixed points. The second section analyses finite cyclic subgroups whose elements have only two fixed points. The third section finds the finite non-diagonal groups in PGL2(K) whose elements have exactly two fixed points. In particular a complete classification is given of the finite subgroups of PGL2(K) when the centre k of K is algebraically closed. Chapter 3 shows that if the centre k of K is algebraically closed, then, any finite subgroup of PGL2(K) is infact conjugate to one in PGL 2(k). It finds the fixed fields in Kk(x) of the finite subgroups of PGL2(k) and shows that their respective generators are the same as in the commutative case.