College Algebra Exam Review 93

College Algebra Exam Review 93 - (c) .a k / l D a kl ....

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2.2. SUBGROUPS AND CYCLIC GROUPS 103 (a) Let R ± denote the rotation matrix R ± D ± cos ± ± sin ± sin ± cos ± ² : Show that the set of R ± , where ± varies through the real num- bers, forms a group under matrix multiplication. In particular, R ± R ² D R ± C ² , and R ± 1 ± D R ± ± . (b) Let J denote the matrix of reflection in the x –axis, J D ± 1 0 0 ± 1 ² : Show JR ± D R ± ± J . (c) Let J ± be the matrix of reflection in the line containing the origin and the point . cos ±; sin ±/ . Compute J ± and show that J ± D R ± JR ± ± D R J: (d) Let R D R ³=2 . Show that the eight matrices f R k J l W 0 ² k ² 3 and 0 ² l ² 1 g form a subgroup of GL .2; R / , isomorphic to the group of sym- metries of the square. 2.2.6. Let S be a subset of a group G , and let S ± 1 denote f s ± 1 W s 2 S g . Show that h S ± 1 i D h S i . In particular, for a 2 G , h a i D h a ± 1 i , so also o.a/ D o.a ± 1 / . 2.2.7. Prove Proposition 2.2.8 . (Refer to Appendix B for a discussion of intersections of arbitrary collections of sets.) 2.2.8. Prove by induction the following facts about powers of elements in a group. For all integers k and l , (a) a k a l D a k C l . (b) .a k / ± 1 D a ± k .
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Unformatted text preview: (c) .a k / l D a kl . (Refer to Appendix C for a discussion of induction and multiple induction.) 2.2.9. Let a be an element of a group. Let n be the least positive integer such that a n D e . Show that e;a;a 2 ;:::;a n ± 1 are all distinct. Conclude that the order of the subgroup generated by a is n . 2.2.10. ˚.14/ is cyclic of order 6. Which elements of ˚.14/ are genera-tors? What is the order of each element of ˚.14/ ? 2.2.11. Show that the order of a cycle in S n is the length of the cycle. For example, the order of .1234/ is 4. What is the order of a product of two disjoint cycles? Begin (of course!) by considering some examples. Note, for instance, that the product of a 2–cycle and a 3–cycle (disjoint) is 6,while the order of the product of two disjoint 2–cycles is 2....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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