Geometry And Collineation Groups of the Finite Projective Plane Pg 22

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Geometry And Collineation Groups of the Finite Projective Plane Pg 22
Ulysses Grant Mitchell
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Suppose n to be less than N/3 . G must then contain m> N/3 collineations of determinant d where d is either i or r. If T be any collineation in G of determinant d 2 the products of the m collineations of determinant d by T are m>N/3 distinct collineations of determi riant unity in G, contrary to supposition. Since n is neither less nor greater than N/3 it follows that n= N/3 The Group of Determinant Unity. By Theorem 5 the group G 60480 has a self-con- jugate subgroup of determinant unity of or...der 60480/3=20160. In §4 it was shown that all collineations of types I a, III, V and those of type I of period 7 and type I of period 5 were of determinant unity. Hence the Group 20160 of determinant unity contains the following collineations: The identical collineation 1 All collineations of type I 3 2240 Those collineations of type Ii which are of period 5, (1-3 of the total number) 8064 Those collineations of type I which are of period 7, (3-10 of the total number) 5760 All collineations of type III 3780 All collineations of type V 315 20160 The group will be designated as above by G 20180 .

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