Let G be set of all 2 × 2 matrices in GL2 (Z5) of the form(²2)(iv) Let N be the subset of all matrices in G of the formnormal subgroup of G.(v) Show that the factor group G/N is cyclic of order 4.(²9)Cwith c, d e Z5 and d 0.with ce Z5. Show that N is a

Answered: Image /qna-images/answer/5d4cf7f5-8f59-417e-aea3-4933b871ccce.jpg

Answered: Let G be set of all 2 x 2 matrices in…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 39E: 39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and ....

See similar textbooks

Similar questions

Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
Prove or disprove that H={ [ 1a01 ]|a } is a normal subgroup of the special linear group SL(2,).
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.
Prove that SL(2,R)={ [ abcd ]|adbc=1 } is a subgroup of GL(2,R), the general linear group of order 2 over R.The subgroup SL(2,R) is called Special linear group of order 2 over R.
39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.
14. Let be an abelian group of order where and are relatively prime. If and , prove that .
let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.
23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .
Let G be the multiplicative group of units U20 consisting of all [a] in 20 that have multiplicative inverses. Find a normal subgroup H of G that has order 2 and construct a multiplication table for G/H.
Let G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.
6. Let be , the general linear group of order over under multiplication. List the elements of the subgroup of for the given, and give. a. b.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

Let G be set of all 2 x 2 matrices in GL2(Z5) of the form (d), (iv) Let N be the subset of all matrices in G of the form normal subgroup of G. (v) Show that the factor group G/N is cyclic of order 4. (19) C with c, d e Z5 and d = 0. with ce Z5. Show that N is a