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Unformatted text preview: ' W G ! H is a bijection that preserves group multiplication (i.e., '.g 1 g 2 / D '.g 1 /'.g 2 / for all g 1 ;g 2 2 G ). For example, the set of eight 3by3 matrices f E;R;R 2 ;R 3 ;A;RA;R 2 A;R 3 A g ; where E is the 3by3 identity matrix, and A D 2 4 1 1 1 3 5 R D 2 4 1 0 1 0 0 0 1 3 5 ; given in Section 1.4 , is a subgroup of GL .3; R / . The map ' W r k a l 7! R k A l ( k 3 , l 1 ) is an isomorphism from the group of symmetries of the square to this group of matrices. Similarly, the set of eight 2by2 matrices f E;R;R 2 ;R 3 ;J;RJ;R 2 J;R 3 J g ; where now E is the 2by2 identity matrix and J D 1 1 R D 1 1 is a subgroup of GL .2; R / , and the map W r k a l 7! R k J l ( k 3 , l 1 ) is an isomorphism from the group of symmetries of the square to this group of matrices. There is a more general concept that is very useful:...
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 Fall '08
 EVERAGE
 Algebra

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