3. Let A and B be groups. Show that S := {(a, e) | a € A} is a normal subgroup ofG := A x B and the quotient group G/S is isomorphic to B.Lot Cha

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Answered: 3. Let A and B be groups. Show that S…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 11E: Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?

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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?
11. Find all normal subgroups of the alternating group .
22. If and are both normal subgroups of , prove that is a normal subgroup of .
Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?
34. Suppose that and are subgroups of the group . Prove that is a subgroup of .
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.
18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
Find all subgroups of the octic group D4.
9. Find all homomorphic images of the octic group.
Find groups H and K such that the following conditions are satisfied: H is a normal subgroup of K. K is a normal subgroup of the octic group. H is not a normal subgroup of the octic group.
Find 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.
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,,,, }.

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3. Let A and B be groups. Show that S := {(a, e) | a € A} is a normal subgroup of G:= A x B and the quotient group G/S is isomorphic to B.