6) Use the given lattices to help find the centralizers of every element in the following groups. (a) D8 Dg (s,r?) (r) (rs,r?) (s) (rs) rs)

Q: In doing plannned comparisons, we use a. The T test for independent groups with MS within in the…

A: Introduction: When there is significant difference between the means of several treatments, the…

Q: 54 The groups An and Sn are not solvable for all n ≥ 5.

A: Solution:-

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

Q: Show that the cyclic group of n objects, Cn, may be represented by r", m = 0, 1, 2,...,n– 1. Here r…

A: Given  Cn is a cyclic group of n elements To show Cn is generated by some element r of Cn. That is…

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

A: Solve

Q: 6g= 3 1 2 3 3 6 * 2 3 2 2 3 ス 3 ス 2 2 3 3

A:

Q: Are groups Z×10and Z×12 isomorphic

A: Concept:

Q: B. Write at least five elements of each of the following cyclic groups. 1. 25Z under multiplication…

A: 25Z under multiplication: {-50, -25, 25, 50, 75}

Q: Give the subgroup diagram for each of the groups: (a) Z24 (b) Z36-

A:

Q: 12. Find the index of (3) in the group Z24.

A: This problem is based on abstract algebra.

Q: In the following Cayley table for a group G, C(3}= 2 3 4 3 4 5 4 1 3 4 1 2 3 2 3 4 4 2 4 O {1,2,4} G…

A: If G be a group and a belongs to G then c(a) = { x : ax = xa , where x belongs in G}.

Q: In the group Zg compute, (a) 6 + 7, and (b) 2-'.

A:

Q: 1. Construct the multiplication table for the the group Us = {1,a, a“, a°, a*} %3D 2mi where a = es.

A: As per the company rule, we are supposed to solve one problem, from a set of multiple problems.…

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

A:

Q: In a group G,let a,b and ab have order 2.show that ab=ba

A:

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

Q: 1. Construct the multiplication table for the the group Us = {1,a, a², a°, aª} where a = 2ni e 5

A: As per our guidelines we are suppose to answer only one ques. Answer of question 1 is as follows:

Q: The cyclic group of order 12 acts on {1,2,..., 12} with the following cycle structure. (1)…

A: Given that,  the group of order 12 In this case of necklace, there is no difference between…

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: The elements of order 8 in the group (Zg ,+) is a)4 b) 5 c)6 d) 8

A: In this question we have to find the number of elements of order 8. (a) 4 is correct option

Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10×Z85 c) Z4xZ25x Z, d) ZnxZ2 x Z65 -21

A:

Q: The following is a Cayley table for a group G. 2*5 4 = 1 1 3 1 4 2 4 4. 4. 5. 4. ENG pe here to…

A: 1

Q: The following is a Cayley table for a group G. The order of 4 * 5 is: 1 3 5 2 3 4 5 2 3 4 2 3 4 5 1…

A:

Q: Q2) If G = Z24 Group a) Is a G=Z24 cyclic? Why b) Find all subgroups of G = Z24 c) Find U,(24)

A: Given that G=ℤ24. a) Then G is generated by the element 1. That is, 1=1,2,3...,22,23,0=ℤ24.…

Q: In the group Z8 compute, (a) 6+7, and (b) 2-1.

A:

Q: 16. A class consist of 15 men and 20 women, in how many ways can a group of 5 be formed if it has to…

A:

Q: True or False? The only simple groups are the groups Zp and An  where p is a prime and n≠4 .

A: False.

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

A:

Q: (a) Draw the lattice of subgroups of Z/6Z. (b) Repeat the above for the group S3.

A:

Q: 4

A: To identify the required cyclic subgroups in the given groups

Q: 5. (a) Ifp is a prime then the group U, has (d) elements of order d for each d dividing p- 1. (b)…

A:

Q: QUESTION 3 Construct the group table for (U(9), ).

A: 3 We have to construct the group table for U9,⋅9. First of all we will write the element of U9,…

Q: ng to a group. If |a| = 12, |6| = 22, and (a) N (b) # {e}, prove that a® = b'1.

A:

Q: 2. What is the order of the element 32 in the group Z36?

A: Modular groups  are cyclic groups. A group G is cyclic if G=<g> for some g in G, where…

Q: (a) How Can we tind thał the groups Give three points and two two not ISomorphic, are examples:-

A: Three points that can be used to prove that two groups are not isomorphic are: Cardinality of the…

Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: Consider the discrete group G of order 8 that has the following Cayley diagram e If we have the…

A: The sequence of operations is fcagec. Each element g of G is assigned a vertex: the vertex set…

Q: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

A:

Q: The group ((123)) is normal in the symmetry group S3 and alternating group A4.

A:

Q: (d) Find the cosets of the quotient group (5)/(10), and determine its order.

A:

Q: In the group U (I6) compute, (a) 5.7, and (b) 3

A:

Q: 2. Let A, B, C, and D be four groups such that A-C and B D. Show that A x B Cx D.

A: We will use the face that A isomorphic to C  and B isomorphic to D to prove the asked result.

Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

A:

Q: The following is a Cayley table for a group G. 2* 5 * 4 = * 1 2 3 4 5 1 2 3 4 5 2 3 4 5 2 3 4 5 2 4…

A: 1

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: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…

A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…

Q: 6.7 Construct a nonabelian group of order 16, and one of order 24.

A:

Q: Let a,b be elements of S6 (symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).verify that…

A: Given: Let a,b be elements of S6 (Symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).

Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,

A: The character table of a group of order 12:

Q: Which of the following is nontrivial proper sub-group of Z4? {0,2}, {0,3}, {0,1}

A: We have given a group Z4 ,  and we have to find which of the following sets {0,2},{0,3},{0,1} is a…