Que 327: Let X, Y be cyclic groups of order 2,3 respectively. Find the orders of the elements of XXY.

Q: An is the alternating group on n letters. What is |An|?

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Q: Fill in the blanks in the table below so that it becomes the Cayley table of a group with elements…

A: We will fill the Cayley table in a way such that it will form a group.

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: In the frieze group F7, write x-3zxyz in the form xnymz k.

A: It is known that F7=xnymzk∥n∈Z,m=0,k=0 or k=1 and x is the translation of R2, and it map u,v to…

Q: How many positive integers n less than or equal to 731 satisfy d(n) = 3?

A: We need to count the number of positive integers n≤731 that satisfy d(n)=3.Here, d(n) is the number…

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

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Q: Ilage 66. 67% 1 0.6 3 22. 2% 2.2 25 80% 0. 88 4 5 354. 55% 0.54 311 53. 33% 7 0.53 15

A: The given data is

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: If a, b, and c are group elements and |a| = 6, |b| = 7, express(a4c-2b4)-1 without using negative…

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Q: Let 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 and B= 2 3 4 5 1 7 8 6 13 8 7 6 5 2 4 Write a, B, and aß as a.…

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Q: The inverse of each element in the group ({1, 3, 5, 7}, ∙8)

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Q: Question: Find the order of every element in the symmetry group of D4. I know the order each group…

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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: How many elements does the group of rotations of a regular hexagon have?

A: The group of rotations of a regular hexagon is D6. And as the order of the group Dn is 2n.

Q: Q2: If G = R- {0} and a * b = 4ab ,show that (G,*) forms a commutative group? %3D

A: To show for the commutative group of (G, *), we verify the following properties of the commutative…

Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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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: Q 5 Find the inverse of the element 9 in the group (U(10), ®10). Q 6 Find the order of the element 2…

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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: 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: Find the inverse of the element 9 in the group (U(10), ®10).

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Q: Give an example of a monoid which is not group. Give an example of a groupoid which is not a…

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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: Find the elements in the alternating groups A3, and A4, respectively.

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Q: Q17 : If a and b are elements in a group G, then the order of ab is the same as the order of ba.…

A: Let us take a non abelian group then their elements will never commute  that is a*b not equal to b*a…

Q: Find all the generators of the following cyclic groups: (Z/6Z,+), ((Z/5Z)*, ·), (2Z, +), ((Z/11Z)*,…

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Q: Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In…

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Q: 1. Let G = Z/36Z. (a) Find all subgroups of G, describe all containments between these subgroups,…

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Q: need help with cyclic groups plz, thanks

A: Given groups are,

Q: Decide which of the elements 2,3, and 5 are generators of the group Z/6

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Q: For G = (Z5 ,+s) , how many generators of the cyclic group G? 5.a O 1.b O 3.c O 4.d O 2.e O

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