College Algebra Exam Review 100

College Algebra Exam Review 100 - ' W G ! H is a bijection...

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110 2. BASIC THEORY OF GROUPS 2.3.8. Consider a regular 10-gon in which alternate vertices are painted red and blue. Show that the symmetry group of the painted 10-gon is isomorphic to D 5 . 2.3.9. Consider a regular 15-gon in which every third vertex is painted red. Show that the symmetry group of the painted 15-gon is isomorphic to D 5 . However, if the vertices are painted with the pattern red, green, blue, red, green, blue, . .., red, green, blue, then the symmetry group is reduced to Z 5 . 2.3.10. Consider a card in the shape of an n -gon, whose two faces (top and bottom) are painted red and green. Show that the symmetry group of the card (including reflections) is isomorphic to D n . 2.3.11. Consider a card in the shape of an n -gon, whose two faces (top and bottom) are indistinguishable. Determine the group of symmetries of the card (including both rotations and reflections). 2.4. Homomorphisms and Isomorphisms We have already introduced the concept of an isomorphism between two groups: An isomorphism
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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 3-by-3 matrices f E;R;R 2 ;R 3 ;A;RA;R 2 A;R 3 A g ; where E is the 3-by-3 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 2-by-2 matrices f E;R;R 2 ;R 3 ;J;RJ;R 2 J;R 3 J g ; where now E is the 2-by-2 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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