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<sub>k</sub>(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, PGL<sub>2</sub>(K). It shows that if K is of suitable characteristic, then any element A of PGL<sub>2</sub>(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 PGL<sub>2</sub>(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 PGL<sub>2</sub>(K) whose elements have exactly two fixed points. In particular a complete classification is given of the finite subgroups of PGL<sub>2</sub>(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 PGL<sub>2</sub>(K) is infact conjugate to one in PGL<sub> 2</sub>(k). It finds the fixed fields in K<sub>k</sub>(x) of the finite subgroups of PGL<sub>2</sub>(k) and shows that their respective generators are the same as in the commutative case.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:457542 |
| Date | January 1979 |
| Creators | Gruza, E. M. |
| Publisher | Royal Holloway, University of London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://repository.royalholloway.ac.uk/items/1deac435-69c6-4013-8744-a371cc599dbe/1/ |
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