Let
In
From Example 2 of section 3.1: Set
Thus we conclude that
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Chapter 3 Solutions
Elements Of Modern Algebra
- Let be a subgroup of a group with . Prove that if and only if .arrow_forward5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forward40. Find subgroups and of the group in example of the section such that the set defined in Exercise is not a subgroup of . From Example of section : andis a set of all permutations defined on . defined in Exercise :arrow_forward
- 42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .arrow_forwardFor each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of G. Sec. 4.6,32 Let a be a fixed element of the group G. According to Exercise 20 of Section 3.5, the mapping ta:GG defined by ta(x)=axa1 is an automorphism of G. Each of these automorphisms ta is called an inner automorphism of G. Prove that the set Inn(G)=taaG forms a normal subgroup of the group of all automorphisms of G.arrow_forward44. Let be a subgroup of a group .For, define the relation by if and only if . Prove that is an equivalence relation on . Let . Find , the equivalence class containing .arrow_forward
- 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 .arrow_forward(See Exercise 26) Let A be an infinite set, and let H be the set of all fS(A) such that f(x)=x for all but a finite number of elements x of A. Prove that H is a subgroup of S(A).arrow_forward43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .arrow_forward
- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.arrow_forward23. Prove that if and are normal subgroups of such that , then for allarrow_forward(See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,