True or False Label each of the following statements as either true or false. 1. Let H be any subgroup of a group G and a Є G. Then aH = Ha implies ah = ha for all h in H. 2. The trivial subgroups {e} and G are both normal subgroups of the group G. 3. The trivial subgroups {e} and G are the only normal subgroups of a nonabelian group G. 4. Let H be a subgroup of a group G. If hH = H = Hh for all hЄ H, then H is normal in G. 5. If a group G contains a normal subgroup, then every subgroup of G must be normal. 6. Let A be a nonempty subset of a group G. Then A = (A). 7. Let A be a nonempty subset of a group G. Then (A) is closed under the group opera- tion if and only if A is closed under the same operation.

Q: 11. Let H be a subgroup of a group G and suppose that 91,92 € G. Prove that the following conditions…

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Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: 1. Let G be a group and H a nonempty subset of G. Then H <G if ab-EH whenever a,bEH

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Q: 22. Let G be a group and let H and K be subgroups of G. For a € G, we define the double coset HaK =…

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

Q: Let G = (Z;, x,) be a group then the inverse of the elements 2, 3 and 6 are O a. 3, 4 and 6 O b. 1,…

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Q: True or False Label each of the following statements as either true or false. ✓1. aHHaØwhere H is…

A: 1. True. For any  a in G , aH and  Ha  both represent the same set as they are both cosets of H with…

Q: Please do the following questions with full handwritten working out.

A: Step 1:

Q: 10. Let G be a group and H a subgroup of G. Let CH) - {gEG|gh = hg. VhEH}, Show that C(H) <G.

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Q: Compute the left regular representation of the group G = {e,a, b} given by the group table below by…

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Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin…

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Q: 4. Let G be a group, and let H be a subgroup of G. Show that if H contains the commutator subgroup…

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Q: Prove or disproe: If C is a groun and a hsC then (ab)-1 h-1a-1

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Q: . Suppose that G be a group and H a subgroup of G. Let g be an element of G. Suppose that the left…

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Q: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: If H and K are subgroups of the group G, then which one of the following is also a subgroup of G? a.…

A: Answer

Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4

A: Given: A Sylow 2-subgroup of S4To find: Describe an isomorphism from this group to D4.

Q: Suppose (G, ◇) is a group. Which of the following is true? 1. ∀a ∈ G, a ◇ e = e ∈ G 2. ∀a ∈ G, a ◇…

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Q: 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…

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Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

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Q: (True or False; Prove or disprove) 1. The order of any subgroup of a finite group divides the order…

A: 1. The order of any subgroup of a finite group divides the order of the group.  This statement is…

Q: Let G be a group. Consider H = {h € G| hg = gh for all g e G}. Then which one of the following is…

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Q: Let G = S4 and N {e, (1, 2) (3, 4), (1, 3) (2, 4), (1, 4) (2,3)}. Show that N is a normal subgroup.…

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Q: Let G = (Z;, x7) be a group then the inverse of the elements 2, 3 and 6 are. O a. 1,2 and 3 O b. 4,…

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Q: 5. Let G be a group and n e Z+ be fixed. Show that H = {a" | a € G} is a subgroup of G

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Q: Prove that the intersection of two subgroups is always a subgroup.

A: In this question, we prove the intersection of the two subgroup of G is also the subgroup of G.