Finite subgroups of PGL₂(K) and their invariants

Abstract

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 K_k(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 K_k(x) of the finite subgroups of PGL₂(k) and shows that their respective generators are the same as in the commutative case.