Decide whether each of the following sets is a subgroup of
a.
b.
c.
d.
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Chapter 3 Solutions
Elements Of Modern Algebra
- Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?arrow_forwardIf H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.arrow_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
- 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_forward18. If is a subgroup of , and is a normal subgroup of , prove that .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
- 34. Suppose that and are subgroups of the group . Prove that is a subgroup of .arrow_forward19. With and as in Exercise 18, prove that is a subgroup of . Exercise18: 18. If is a subgroup of , and is a normal subgroup of , prove that .arrow_forwardLet be a group of order 24. If is a subgroup of , what are all the possible orders of ?arrow_forward
- 19. Prove that each of the following subsets of is a subgroup of . a. b.arrow_forwardFind subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.arrow_forwardLet H be the subgroup (1),(1,2) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,