Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

Q: In (Z13, +13) the set K = {0,3,5,8,9,11} is eyclie subgroup. True False

A: Use the definition and properties of a cyclic group. Check whether the whole group can be generated…

Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: The Galois group of  x4 +x2 +1 is the same as that  of  x6 -1 and is of order 2.

A: Since the primitive nth root of unity is        e2πi/n=cos2πn+i sin2πn. Then primitive third root…

Q: Consider the groups U(8)  and Z4 . (i)  Determine the identity element in the group U(8)xZ4 . (ii)…

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Q: Let G be a group with identity element e. Prove that for any element a in G, (a^2)^-1 = (a^-1)^2.…

A: We are given that G is a group with an identity element e. In group theory, a branch of mathematics,…

Q: Consider the group G = Z4 X Z9 where the operation is done componentwise, that is (a, b) * (c,d) =…

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Q: The Galois group of  x4 +x2 +1 is the same as that  of  x6 -1 and is of order 2.

A: Since the primitive nth root of unity is        e2πi/n=cos2πn+i sin2πn. Then primitive third root…

Q: 3. In each part, find the order of the given element in the given group. (a) i= √-1 in the…

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Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

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Q: Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) =…

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Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

Q: find multiplication table of dihedral group D_4where rotations are denoted by r,r^2, r^3.…

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Q: Calculate the cyclic subgroup (15) < (Z24, +21)

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Q: 4. There is a theorem that says that every element g € GL(2, R) can be written, in a unique way, as…

A: Given:  g=053-12           We have to find k, a, n such that g=kan where k∈K, a∈A and n∈N.

Q: Find the order of the given factor group (Z3 x Z6)/

A: If G be a group, and H be a subgroup of G, then the order of the factor group G/H is given by…

Q: 32) Prove that every subgroup of Q8 in normal. For each subgroup, find the isomorphism type of its…

A: The elements of the group are given by, Q8=1,-1, i, -i, j,-j, k, -k Note that every element of Q8…

Q: Write a Cayley table for the dihedral group D6 = {1, r,r^2,s,sr,sr^2|

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Q: Determine two non trivial subgroups of U(18)

A: This question is related to Abstract Algebra

Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: 12. (Exercise 16.22) Prove the third isomorphism theorem. At each major step in the proof, explain…

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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: Example: Show that (Z,+) is a semi-group with identity

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Q: Q1// Let H={2^n: n in Z}. Is H subgroup of Q- * {0}

A: Given the set H = { 2n | n lies in Z } we have to prove that ( H, × ) forms a subgroup of ( Q - {0},…

Q: Draw the lattice diagram of subgroups of S3 and the corresponding lattice diagram of subgroups of…

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Q: Find all subgroups of the group Z36 and draw the subgroup diagram for the subgroups.

A: Given group is We have to find all subgroups of the group And also we have to draw its subgroup…

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: List all elements of U(10) and give a multiplication table for the group U(10)

A: Given  that, The group U(10). We have to find the all elements in U(10) and multiplication table for…

Q: Consider the group G={1,−1,i,−i} under multiplication, where i^2=−1. Determine the inverse of each…

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Q: n point group D2d, for each of the irreducible representations, verify that the sum of the squares…

A: In point group D2d portions of the D2d character table D2d E 2S4 C2 2C2' 2□d A1 1 1 1 1 1 A2…

Q: On the in-class portion of Exam 1, you saw that Z5 - {[0]} is a group under multiplication. Find all…

A: There are three distinct cyclic subgroups of Z5x​​;{1}{1, 2, 3, 4} (which is the entire group){1,…

Q: D3) A\ give an example of a non-cyclic group m wineih every proper subgroup is cyclic ( explain your…

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Q: give the order of the element in the factor group (2,0) + in (Z6 x Z8)/

A: Let    G be a group, and let    g be an element of that group.The order of the element   g, denoted…