4*. Let a and b be elements of orders 5 and 8 in a group of order 120. List the possible orders of the subgroup (a, b). (Defend your answer.)

Q: Construct the group table for (Z,, + ,).

A: We have to construct the group table for (Z5,+5). Where Z5 is modulo 5 group of integers. So to…

Q: - Show that the following subset is a subgroup. H = {o e S, l0(n) = n} S,

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Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345

A: For group, 2*3*4=(2*3)*4.

Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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Q: 28. Is every group a cyclic? Why?

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

A: According to our company's guidelines I can only answer first question since you have asked multiple…

Q: %24 W NI 52 The following is a Cayley table for a group G2 3 4 12 12 3 2 1 31 4 4 3 O 2

A: To find the 2*3*4

Q: o:Z, →Z, A GROUP HOMOMORPHISM 30 30 KER () = {0,5,10,15, 20, 25} o(13) = 6 now THAT, 0" (6) FIND ?

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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: Is {0, 2, 4, 6} a subgroup of Zg? Justify your answer.

A:

Q: 11. Prove that every Cayley table is a Latin square for a group. That is, each element of the group…

A: To prove, each element of the group appears exactly once in each row and each column of a Cayley…

Q: If we have 6 subgroups as 36.2,37.0,39.6 41.02,44.3 , and 45.5. Graph Z and MW charts and plot the…

A: A z-mR control chart is two charts.  One chart is for the z value which is the “standardized” X…

Q: 25. o: Z4 → Z12

A: Homomorphism : Let us consider a map f: V→W then f is said to be homomorphism if for all v,u∈V…

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: 10. Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not…

A: Given that,   (a) Group of order 40. Let G=40,  i.e, G=40=23.5, Since, 5-Sylow subgroup, i.e…

Q: The following is a Cayley table for a group G. The order of4 is: 2 1 3 3 4 1 4 1 4 1 5. 2 512 m 45…

A: By observing the Cayley table :

Q: Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not simple.

A: To show that the group of order 45 is not simple.

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

A:

Q: What would a non-transitive graph of three groups, lets say (A, B, and C) look like?

A: Given non transitive graph of three groups

Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

Q: 10) Which of the following is a group? * O (Z,*), a* b = a + b - 1 va, b e Z O (Z,*), a* b = a - 2b…

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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: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

A: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

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…

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Q: Suppose that G is a cyclic group and that 6 divides |G|. How manyelements of order 6 does G have? If…

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Q: 4-Let (Z12, +12) be a group and let S={4,6}, find subgroup H generated by S. if exist

A: I have used the definition of subgroup generated by a subset.

Q: 13. Find groups that contain elements a and b such that |al = \b| = 2 and a. Jab| = 3, b. Jab| = 4,…

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Q: If G is an infinite group, what can you say about the number ofelements of order 8 in the group?…

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Q: Let Ø:Z50→Z15 be a group homomorphism with Ø(x)=7x. Then, Ker(Ø)= * O {0, 10, 20, 30, 40} None of…

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Q: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

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Q: Draw the lattice of the subgroups Z/20Z.

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

A:

Q: Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than 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: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

A:

Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

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Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 7x. Then, Ø-'(7) : O {0, 15, 30, 45} O {1, 6,…

A:

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: (b) Consider the incomplete character table for a group given below:

A: The table for the incomplete character is,   1 a b c d x1 1 1 1 1 1 x2 1 1 -1 -1 1 x3 1 1…

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

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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…

Q: In Z24 the number of all subgroups is 8 O 6.

A: We are asked to find the subgroup of z24