2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a) -{(82) A = d>0} a> 0 and d >
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- Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:9. Suppose that and are subgroups of the abelian group such that . Prove that .
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .Find two groups of order 6 that are not isomorphic.
- Let be a subgroup of a group with . Prove that if and only if .Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.