1. State, with reasons, which of the following statements are true and which are false. (a) The dihedral group D6 has exactly six sub groups of order 2. (b) If F is a free group on two generators x and y, and F₁CF is a subgroup that does not contain x or y. Then F₁ is not isomorphic to F.

Q: The doctor prescribes 25mg/kg divided into 3 equal doses for a todddler whom was 42 lbs. find the…

A: The weight of the child is 42 lbs The weight of the child in kilograms is 42×0.4536=19.0512≈19…

Q: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

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Q: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

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Q: 10. If : G→ H is a group homomorphism and G is cyclic, prove that (G) is also cyclic.

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Q: 3. Prove that the groups (R.) and (R,+) are not isomorphic, where (R* R{0},-) is the multiplicative…

A: We have to show that the groups ℝ, + and ℝ*, · are not isomorphic. Here, the group ℝ*, · does not…

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Q: 1. Prove or disprove that the given set is a subgroup of the given group G. (a) H = {a + bi|a,b e R,…

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Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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Q: 5. Let G be a group. (a) Prove that Inn(G) is a subgroup of Aut (G) (b) Prove that for any € Aut(G)…

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Q: 1) Let G = {e,a,a',a²,a*,a,a°,a'] where a = e and let H = (a*) be a cyclic subgroup %3D %3D %3D of…

A: We have to find quotient group  G/H .

Q: 3. Let a, b, c be three distinct positive real numbers, and let {(x, y, z) = R³ ||x| ≤ a, │y| ≤ b,…

A: Given that:Let  be the three distinct positive real numbers, and let B be the subset of .a) To Find…

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…

Q: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

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Q: 4. Let G = {z € C|z¹ = 1 for some n € Z+} Prove that G is a group under multiplication, but not…

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Q: Which of the following sets forms a subgroup in the additive group Z? (a) {1} (b) N (c) {1, -1}. (d)…

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Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

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Q: Find a matrix M such that M-¹0 (8) M = (19) for every element g € S3. Conclude that the…

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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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: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

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Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: Let G be a group acting on a set S. Let x and y be elements in S such is in the orbit of x under the…

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Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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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: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

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: Find the partition of Ze into cosets of the subgroup H = {0,3}.

A: In this problem we have to find the partition of  into cosets of the subgroup .