[1 a {[69] : a € R}. G is a group under the operation of matrix multiplication 01 2. Let G := (you do not need to show that G is a group). Prove that G is isomorphic to R.
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- 5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.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.
- 24. Let be a group and its center. Prove or disprove that if is in, then and are in.31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.Label each of the following statements as either true or false. Let x,y, and z be elements of a group G. Then (xyz)1=x1y1z1.
- True or False Label each of the following statements as either true or false. 10. The nonzero elements of form a group with respect to matrix multiplication.True 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.Let be a subgroup of a group with . Prove that if and only if