In the group Z24, let H =(4) and N= (6). (a) State the Second Isomorphism Theorem. (b) List the elements in HN (which we might write H+ N for these additive groups) and in HON. (c) List the cosets in HN/N, showing the elements in each coset.

Q: In the group R* find elements a and b such that lal = ∞, lbl = ∞ and labl 2.

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Q: -) Write the Cayley table for the group U(15).

A: U(15) is the set of all elements in ℤ15={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} ; x≡x ,such that…

Q: Let G - { (a, b) : a eR, be R\{0} }. Define on G the binary operation: (a, b) (c , d)- (a + bc , bd)…

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Q: If x is an element of a cyclic group of order 15 and exactly two of x3, x5, and x9 are equal,…

A: Given: The order of group is 15

Q: (H,*) is called a of (G,*) if (H,*) is a group.

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Q: Find the regular permutation group isomorphic to the multiplicative group G={1,-1,i,-i}.

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Q: For each integer n > 1, give examples of two nonisomorphic groupsof order n2.

A: To Do: For each integer n > 1, give examples of two non-isomorphic groups of order n2.

Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: means a cyclic group but 3+i can't turn into 3-i with a positive integer exponent.

A: Please see the answer in next step.

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: Determine the conjugacy classes of Ag and the class equation for Ag.

A: Conjugacy relation:  Let G be a group and ~be a relation on group G given by b~a if and only if b is…

Q: 4. The group defined by the following table is called the group of quaternions. Use the table to…

A: NOTE:- According to the company guidelines we are supposed to answer only first 3 sub-parts. Kindly…

Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

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Q: Express the following as products of cyclic gr a) Z140 ») U (140) =) Z400 H) U (400).

A: To express the given groups as products of cyclic groups, we need to find the prime factorization of…

Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

Q: Find all homomorphic images of dihedral group, D3.

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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: Consider the group G = ℤ9 ⨉ ℤ27 where the operation is done componentwise, that is (a, b) * (c, d)…

A: The given Group is ,                   G = ℤ9×ℤ27 The operation is componentwise , i.e.         (a ,…

Q: (3) If H={0,6,12,18}, show that (H,+24) is a cyclic subgroupof (Z4,+24). Also list the elements of…

A: Subgroup

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

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Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.

A: Given,ab2=a2b2To prove: ab=ba

Q: Let G= be a cyclic group of order 60, and H=. List all left cosets of H in G.

A: Let G be a group and H be a subgroup of G. We define the left co-set of H by a as aH = ah : h∈H Then…

Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: Does (U(14),×14) form a cyclic group? If yes, find all generators.

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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: which of the following group are cycle ?(a) U(35) (b)U5(40)

A: It is required to check which of the given group are cyclic. (a) U(35) (b)U5(40)

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: (2) (Z x Z)/(6Z × 14Z)

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

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Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

Q: (1) Let C₁ = denote the cyclic group of order n. (a) a. Show that C₂ C₁ is a cyclic group. (b) Show…

A: Direct sum of the cyclic group

Q: In the group Z24, let H =(4) and N=(6). (a) State the Second Isomorphism Theorem. (b) List the…

A: Given that, Group is  Z24, let H=4 and N=6. Let (G,∗) be a group, H a subgroup of G, and g∈G. The…

Q: Up to isomorphism, list all abelian groups of order 1350.

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Q: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

Q: Find the regular permutation group isomorphic to the multiplicative group G = {1, −1, i, −i}.

A: We will find the regular permutation group isomorphic to the multiplicative group…

Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

A: