Alg - Groups and Geometry The Second Part of Algebra and...

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Unformatted text preview: Groups and Geometry The Second Part of Algebra and Geometry T.W.K¨orner April 20, 2007 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). What is presented here contains some results which it would not, in my opinion, be fair to set as book-work although they could well appear as problems. (A book-work question asks you for material which you are supposed to know. A problem question asks you to use known material to solve a question which you may not have seen before. A typical Cambridge examination question consists of book-work followed by a problem, the so-called ‘rider’, which makes use of the book-work.) I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. These notes are written in L A T E X2 ε and should be available in tex, ps, pdf and dvi format from my home page http://www.dpmms.cam.ac.uk/˜twk/ My e-mail address is twk@dpmms . Supervisors can obtain notes on the exercises in the last four sections from the DPMMS secretaries or by e-mailing me for the dvi file. Contents 1 Preliminary remarks 2 2 Eigenvectors and eigenvalues 3 3 Computation 6 4 Distance-preserving linear maps 9 5 Real symmetric matrices 13 1 6 Concrete groups 17 7 Abstract groups and isomorphism 21 8 Orbits and suchlike 25 9 Lagrange’s theorem 28 10 A brief look at quotient groups 30 11 The M¨obius group 33 12 Permutation groups 37 13 Trailers 40 14 Books 45 15 First exercise set 46 16 Second exercise set 52 17 Third exercise set 57 18 Fourth exercise set 61 1 Preliminary remarks Convention 1.1. We shall write F to mean either C or R . Our motto in this course is ‘linear maps for understanding, matrices for computation’. We recall some definitions and theorems from earlier on. Definition 1.2. Let α : F n → F n be linear and let e 1 , e 2 , ..., e n be a basis. Then the matrix A = ( a ij ) of α with respect to this basis is given by the rule α ( e j ) = n summationdisplay i =1 a ij e i . We observe that, if x = ∑ n j =1 x j e j and α ( x ) = y = ∑ n i =1 y i e i , then y i = n summationdisplay j =1 a ij x j . 2 Thus coordinates and bases go opposite ways. The definition chosen is con- ventional but represents a universal convention and must be learnt. Theorem 1.3. (Change of basis.) Let α : F n → F n be a linear map. If α has matrix A = ( a ij ) with respect to a basis e 1 , e 2 , ..., e n and matrix B = ( b ij ) with respect to a basis f 1 , f 2 , ..., f n , then there is an invertible n × n matrix P such that B = P − 1 AP. The matrix P = ( p ij ) is given by the rule f j = n summationdisplay i =1 p ij e i ....
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Alg - Groups and Geometry The Second Part of Algebra and...

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