12.3.1. Let G be a group of order 585 = 3² × 5 × 13. (a) Prove that G is not simple. The group G continues to have 585 elements. Let S = Syl3(G) be the set of Sylow 3-subgroups of G. Let PE S. Assume that P is not normal in G. (b) What can you say about |S|? (c) What can you say about |NG(P)|?

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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Q: = (1) Let G be a group of order 150 = 2 × 3 × 52. Assume N G with |N| = 6. Prove that N is the only.…

A: Proof by Contradiction: 1.Assume the Opposite: Let's assume there exists another subgroup of order 6…

Q: Prove that a group of order 28 must have an element of order 2.

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Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

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Q: Go Let G be a group and a, b = G. Prove that (ab)−¹ = b¯¹a-¹. Let G be a group. Prove that for all…

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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: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

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Q: Q1) Consider the group Z10X S5. Let g = (2, (345)) € Z10X S5. Find o(g). T LOV

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: 3. Give the lattice of subgroups of Z₂ X Z₁.

A: Sol:- The group Z2 × Z4 is the direct product of two cyclic groups of order 2 and 4, respectively.…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

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Q: 10. Let G be a finite group and a € G. Use Lagrange's Theorem to prove that the order of a divides…

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Q: . Let H (27) be the subgroup of G=Z36. What is the order of the quotient group G/H? = (a) 3 (b) 6…

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Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: 18. Let G be the group (Z/7Z)x. Determine all the cyclic subgroups of G.

A: In this context,  represents the multiplicative group of integers modulo 7 that are relatively…

Q: What is the order of the group  ℤ10 x  ℤ15 x U(10) /〈(9, 10, 7)〉?

A: To determine the order of the group ℤ10 × ℤ15 × U(10) / 〈(9, 10, 7)〉, we need to divide the order of…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

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Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: Q4: Let A and B be subgroups of a group to that A1 = 4 and 1B1-5- Show that AnB = {e}

A: Let A and B be the two subgroups of Group GSuch  thati.e., the Order of subgroup A is 4 and B is 5…

Q: 2.1.12. Corollary Let (H)be a nomal subgroup of the group (G). There is one -one correspondence…

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Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: (7.1) Let (G,) be a group. Prove that the right cancellation law holds in G. Provide reasons in your…

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

Q: 4. Let (G, *) be a group of order 231 = 3 × 7 × 11 and H€ Syl₁₁(G), KE Syl, (G). Prove that (a). HG…

A: The Sylow theorems are a fixed of theorems named after the Norwegian mathematician Peter Ludwig…

Q: (1) Find all subgroupsof (Zs.+s).

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 2. Let G = Z₂ x Z₁. (a) Find subgroups H and K of G such that HK but G/H/G/K. (b) Find subgroups A…

A: (a) Here, G = Z2 × Z4                = { (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1,…