Let (G, 0) be a group and x E G. Suppose H is a subgroup of G that contains x. Which of the following must H also contain? CALL "powers" x0x, x0x 0x,... OThe identity element e of G All elements xyfory EG Ox, the inverse of x Enter the smallest subgroup of Z13* containing the element [9]13, as a set. Write each congruence class in the form [b]13 where 0≤b< 13.

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:

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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…

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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 H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

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Q: 2. Let G and H be groups and 4 : G → H be a homomorphism. a. The kernel of p is the set Ker o = {g €…

A: The solution to the given questions is explained below.

Q: 9. Let (G,*) be a group for which H is a subgroup. For any element aɛG. The following set a + H =…

A: Since the left coset of H in G is defined by a+H=a+h  h∈H, the group G is an additive group.  We…

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Q: Let G be a group and let a e G. The set C(a) = {x € G | xa = that commute with a is called the…

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Q: 5. If the pair of cycles a and ß have no entries in common, then aß= ßa. True False ●

A: answer is in next step

Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…

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Q: Show that if H₁ is a subgroup of G₁ and H₂ is a subgroup of G2, then H₁ × H₂ is a subgroup of G₁ ×…

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Q: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

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Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

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Q: Let K and H be subgroups of a finite group G with KCHCG.If[G:H] = 4 and [H:K] = 3. Then, [G:K] = 3 4…

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Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Determine which of the following are groups under the stated operations. For those which are groups,…

A: i) the pair here we draw composition table :246824826486426246886284from above table ,we concluded…

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: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

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Q: 4. Let G and G be groups and let y be an isomorphism from G to G. Prove that if KCG, then y(K) CG.

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Let (G, 0) be a group and x € G. Suppose H is a subgroup of G that contains x. Which of the…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

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Q: Let G be a group and a 2 G. Then the set = {a" : ne Z}is a subgroup of G. 48.

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Q: Let (G, 0) be a group and x = G. Suppose H is a subgroup of G that contains x. Which of the…

A: Let G, * be a group and x∈G. Suppose H is a subgroup and it contains the element x. Find out which…