Let
(This mapping is called the trace of the matrix.) Prove or disprove that
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Elements Of Modern Algebra
- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forward11. Let be , and let be the group of nonzero real numbers under multiplication. Prove that the mapping defined by Is a homomorphism, and find ker . Is an epimorphism? Is a monomorphism? (The value of this mapping is called the determinant of the matrix.)arrow_forward39. 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.arrow_forward
- 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.arrow_forward2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .arrow_forward9. Find all homomorphic images of the octic group.arrow_forward
- 15. Repeat Exercise with, the multiplicative group of matrices in Exercise of Section. 14. Let be the multiplicative group of matrices in Exercise of Section, let under multiplication, and define by a. Assume that is an epimorphism, and find the elements of. b. Write out the distinct elements of. c. Let be the isomorphism described in the proof of Theorem, and write out the values of.arrow_forwardWrite 20 as the direct sum of two of its nontrivial subgroups.arrow_forwardTrue or False Label each of the following statements as either true or false. 11. The invertible elements of form an abelian group with respect to matrix multiplication.arrow_forward
- Let G=I2,R,R2,R3,H,D,V,T be the multiplicative group of matrices in Exercise 36 of Section 3.1, let G=1,1 under multiplication, and define :GG by ([ abcd ])=adbc a. Assume that is an epimorphism, and find the elements of K=ker. b. Write out the distinct elements of G/K. c. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of . Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G.arrow_forwardFind all subgroups of the quaternion group.arrow_forwardTrue or False Label each of the following statements as either true or false. 9. The nonzero elements of form a group with respect to matrix multiplication.arrow_forward
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