What is the method used in the solution part that is highlighted in the answer? If possible, kindly give some references to it. (Given) Let G be a finite group and H1, H2, ..., Hk be subgroups of G.(Question) If H;< Hj, show that [G: H;] = [G : H;] [Hj : H;].(Answer) We know that [G : H;]|G||Hi|Given Hi< Hj →Hj N Hị = HịSolution:=IG||Hj n Hi||Hi|[G: H;]|G||G||Hj] _ |G||Hj|[G: H;] [Hj : H;]|Hj n Hi||Hj||Hj||Hi|

Answered: Image /qna-images/answer/e0ced523-a3e2-40e9-94b5-c828fff57a71.jpg

Answered: (Given) Let G be a finite group and H1,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 24E

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Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup (1),(2,3) 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. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.
13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such that , prove that .
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.
24. Let be a group and its center. Prove or disprove that if is in, then and are in.
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34. Suppose that and are subgroups of the group . Prove that is a subgroup of .
True or False Label each of the following statements as either true or false. Let H be a subgroup of a group G. If hH=Hh for all hH, then H is normal in G.
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