G be a group and let a e G have order 30. List the powers of a that have order 2, the rers that have order 3, and the powers that have order 5.

Q: What is the minimum number of generators needed for Z2 ⨁ Z2 ⨁ Z2?Find a set of generators and…

A: Given: Group, Z2⊕Z2⊕Z2To find: The minimum number of generators and a set of generators and…

Q: Which of the following groups are solvable groups: Z/nZ, D8, and S4.

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Q: Prove that a group of order 105 contains a subgroup of order 35.

A: Let G be a group of order 105. We need to show that G contains a subgroup of order 35. We know the…

Q: Let a be an element of order 8 in a group G. 6 Then order of a Select one: O a. 6 о Б.48 C. 2 d. 4

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Q: Give examples of four groups of order 12, no two of which are isomorphic.Give reasons why no two are…

A: We have to give 4 examples of order 12, such that all are non-isomorphic to each other.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: Let G be a cyclic group of order 135. The number of generators of G is The number of elements of…

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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: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: 1. Show that every group of prime order is simple.

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Q: 4*. Let a and b be elements of orders 5 and 8 in a group of order 120. List the possible orders of…

A: To list the possible orders of the subgroup generated by a and b under the given conditions

Q: 4.6 List all the elements of (Z45,) that are of order 15.

A: (.) Given  group is  ⊕ .(.). The group  ⊕ is an abelian group under addition modulo   and given by ,…

Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

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Q: (S) Is all groups of ve, glve aIl pie. about groups of order 5? (Are they always commutative).

A: Concept:

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

A: G be a group of order 3 Consider G=e,a,bwhere e is the identity then the Cayley table is…

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 and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: 4. Let G be a group and g E G be an element of order 8. Find the order of the following elements.…

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