2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

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: Let G = <x, y | x2n = e, xn = y2, y-1xy = x-1>. Show that Z(G) ={e, xn}. Assuming that |G| =…

A: To show that Z(G) = {e, xn}, we need to find all elements in G that commute with every other element…

Q: group. What do you mean by conjugate elements and classes? Find the classes of the (Rohilkhand 1999)…

A: In the given problem, the definition of 'conjugate' elements and classes in a group are used. Find…

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: Consider the groups Z[i] = {a + bi | a, b = Z, i² = -1} and Z (both under +), and the function o:…

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Q: For G being each of the following groups, compute the centre Z(G): (a) S3 (b)Z/3Z x S3

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Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: 3. Let Dn (n ≥ 3) be the dihedral group of order 2n. (i) Show that D10 ≈ D5 × Z2 by constructing an…

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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…

Q: Suppose that for the three elements x, u, v of a group G, x=11v=vu, u²=e, v9=e where p and q are…

A: Given That : three element x, u ,v of a group G, x=uv=vu,up=e,vq=e where p and q are relative prime…

Q: (a) (i) Let G = R-{0} × R. Prove that (G, *) is a group where is defined by (a, b) * (c, d) = (ac,…

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

Q: Consider the groups (R,+) and (R x R. +). Define the map Rx (RX R) → Rx R defined by ra (z. y) =…

A: The given groups are ℝ, + and ℝ×ℝ, +. The map ↦: ℝ×ℝ×ℝ→ℝ×ℝ is defined as, r↦x, y=x+ry, y Definition:…

Q: 7. [10] Let G be a group, with x € G and o (x) = 48. Give the order of the following elements of G:…

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Q: Neutral element (Identity element) in the group {(Q\0),.} is   A 1   B -1   C Does not exist…

A: Given :- (Q\0) is a group under usual multiplication.  To find :- Neutral element (Identity element)…

Q: find the fundamental group of X := {(x, y, z) = R³|(x² + y²) (y² + z²)(x² + z² − 1) = 0}

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Q: Consider the group A5. (a) Compute a = (1,2, 3)(3, 4, 5) and ß = (1, 2, 3)(2, 3, 4). (b) List…

A: Solution given below

Q: 1. Prove or disprove (a). The order of the element (2, 3) € Z6X Z15 is 5. (b). The group Z7X Z17 x…

A: As per the guidelines, I am supposed to do the first three sub parts. Kindly post the other…

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Let X, Y be cyclic groups of order 2, 3 respectively. Find the orders of the elements of XxY.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: Consider the group (Zn,+). Find the center of Zn (ie. find Z (Zn)).

A: Center of the given group is Zn itself.Explanation:Step 1: Step 2: Step 3: Step 4:

Q: 51. Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove…

A: According to our guidelines we can answer only first question and rest can be reposted. Not more…

Q: Calculate G/H for G = V, the Klein’s four-group and H = (b).

A: The Klein four-group is defined by the group presentation V=a, b| a2=b2=ab=c2=e Given: G=e, a, b, c…

Q: Q5. Let A and B be two groups. Let 0: A x B → B defined by 0(a, b) = b Is 0 isomorphism? Find…

A: To check whether a function θ is isomorphism, it is required to check  θ is homorphism θ is one-one…

Q: How would you solve such a question: Suppose f : Z → Z5 is a group homomorphism, and suppose f (3) =…

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Q: . Let ϕ : Z31 → Z31 be a homomorphism such that ϕ(16) = 19. Calculate ϕ(1) and ϕ(10).

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Q: 3. Infinite, abelian (and additive), and so that for every element a, a+a=0. (Hint: Think of…

A: Note the question asks us for an example of an infinite group such that sum of two elements is…

Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

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

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

Q: (a) Make addition and multiplication tables for Z/3Z.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: 64. Express Ug(72)and U4(300)as an external direct product of cyclic groups of the form Zp

A: see my attachments

Q: 2. There is an element from the group Qu of order: (a)3 (b) 5 (c) 6 (d) 4 The correct answer is (a)…

A: We know order of an element in a divides the order of the group. We know center of Sn is trivial for…

Q: Let α = (12)(34) and β = (24). Show that the group generated by αand β is isomorphic to D4.

A: It is given that α=1234 and β=24. It is required to show that the group generated by α and β is…

Q: For each of the following groups, find the order of the group and the order of each element in the…

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Q: A All B (ii), (iii) D E None (i), (iii) (i), (ii) Suppose such that (i): (ii): : Z→ Z40 is a group…

A: Please find attachment

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: 3. Show that G = {a+bv2 |a,b e Q} is a group under the usual addition.

A: 3. Let G is said to be group under binary operation (.). It should satisfy the following properties.…

Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: 2. Let G = (1, 0). Decide if G is a group with respect to the operation * defined as follows: x * Y…

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Q: Let Dn be the dihedral group of order 2n. Decide whether the following pairs of groups are…

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Q: Write out the Cayley table for the dihedral group D6 = {1,r,r2,s,sr,sr2}. (In general, we apply the…

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Q: Show that the group (U7, *) is isomorphic to {Z6, +}. how that there is an isomorphism between the…

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Q: Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload…

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Q: 1. There can only be one group of order h, when h is a prime number (T) or_(F) 2. There is only one…

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

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: Find the order of the element (2, 3) in the direct product group Z4 × 28. Compute the exponent and…

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Q: Consider the group GL₂(R). Let A (a) Find all elements of H = (A). (b) What is A|? √2 2 √2 2 NANS √2

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Q: 4. (a) (8 points) Prove the following two groups of order 8 are not isomorphic: Z8 Z4 × Z2.

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