Let G be the set consisting of the following matrices: (6 9) (6 ) ( ) -1/2 v3/2 V3/2 1/2 -1/2 V3/2) -1/2 -1/2 V3/2) -V3/2 1/: V3/2 -1/2 Show that G forms a group under multiplication, and write the Cay- ley table for this group.
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- Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.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.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.
- True 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.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.Find two groups of order 6 that are not isomorphic.
- 40. Prove or disprove that the set in Exercise is a group with respect to addition. 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.9. Suppose that and are subgroups of the abelian group such that . Prove that .Label each of the following statements as either true or false. The Cayley table for a group will always be symmetric with respect to the diagonal from upper left to lower right.