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1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of GL(n, R). car

1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of GL(n, R). car

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 33E
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1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of GL(n, R). 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) (b) 3. a :{(89) 0 A = : a>0 and d > 0 :>0} N = = {(1 i): bER} Here's a trickier example of a subgroup of GL(2, R): *)}~{ ( K = cos sin 0 0 5 3 -12 - sin cos Prove that K is indeed a subgroup of GL(2, R). (You will probably recognize the elements of K from an earlier homework.) cos o sin o sin o -3 -17 3 5)} 4. There is a theorem that says that every element g E GL(2, R) can be written, in a unique way, as kan for some k EK, a E A, and n E N (with K, A, N as in the last two problems). Your job: (a) If g = - cos o find k, a, n, such that g = kan. (b) If g = find k, a, n, such that g = kan. For both of these, show your work and explain how you found your answers. Helpful fact: if det g> 0, then k will be a rotation, and if det g < 0, then k will be a reflection. 4. Find all the element of Z/8 that have order 8. Find all the elements of Z/9 that have order 9. Find all the elements of Z/72 that have order 72. (Fun fact: there is a relationship
Transcribed Image Text:1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of GL(n, R). 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) (b) 3. a :{(89) 0 A = : a>0 and d > 0 :>0} N = = {(1 i): bER} Here's a trickier example of a subgroup of GL(2, R): *)}~{ ( K = cos sin 0 0 5 3 -12 - sin cos Prove that K is indeed a subgroup of GL(2, R). (You will probably recognize the elements of K from an earlier homework.) cos o sin o sin o -3 -17 3 5)} 4. There is a theorem that says that every element g E GL(2, R) can be written, in a unique way, as kan for some k EK, a E A, and n E N (with K, A, N as in the last two problems). Your job: (a) If g = - cos o find k, a, n, such that g = kan. (b) If g = find k, a, n, such that g = kan. For both of these, show your work and explain how you found your answers. Helpful fact: if det g> 0, then k will be a rotation, and if det g < 0, then k will be a reflection. 4. Find all the element of Z/8 that have order 8. Find all the elements of Z/9 that have order 9. Find all the elements of Z/72 that have order 72. (Fun fact: there is a relationship
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