22. Let G be a group and let H and K be subgroups of G. For a € G,we define the double coset HaK = {hak: he H, ke K}. Prove thatif a, b E G and HaKnHbK #0, then HaK = HbK.

We show the equality of the sets by showing one is contained in the other and vice-versa.

Answered: 22. Let G be a group and let H and K be…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 18E: 18. If is a subgroup of the group such that for all left cosets and of in, prove that is...

See similar textbooks

Similar questions

45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
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?
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,,,, }.
If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
Let be a subgroup of a group with . Prove that if and only if
9. Find all homomorphic images of the octic group.
24. Let be a group and its center. Prove or disprove that if is in, then and are in.
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.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

22. Let G be a group and let H and K be subgroups of G. For a € G, we define the double coset HaK = {hak : h = H, k € K}. Prove that if a, b E G and HaK ~ HbK ‡ Ø, then HaK = HbK.