Let a and b belong to a group. If la] = 24 and |b| = 10, what are the possibilities for (a) n (b)i?

Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is

A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…

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: If a, b are elements in a group G, show that (ab)-1= b-1a-1

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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: 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: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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

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Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

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: Let G be the set of nonnegative integers. Is a group, |a - b| for all a, b, EG? where a * b =

A: We will check whether it is group or not.

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

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Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

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Q: 2) (b*a)1 = . If a, b are elements of a group G?

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Q: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is     a 1   b -1…

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

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Q: Let a and b belong to a group. If la| = 12, \b| = 22, and (a) N (b) + {e}, prove that a6 = bl1.

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

A: using  third Sylow Theorem

Q: If a,b are elements in a group and a-2, b-3. them Ka)N(b=6

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Q: 10. Given the group (S = {a,b,c, d},®) a a d ic a b a a d d The inverse of an element r, x E Sis D-…

A: Inverse element is an element which when operated with its inverse gives the identity

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: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

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. 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: Prove that in a group, (ab)^2=a^2b^2 if and only if ab=ba.

A: Proof:Let a,b ∈ G.Assume (ab)2 = a2b2 and that prove ab = ba as follows.

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: 27. If g and h have orders 15 and 16 respectively in a group G, what is the order of (9) n (h)?

A:

Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: is a group with identity (eg, eH).

A:

Q: The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1 for a, beZ, is a group.…

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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Q: Prove that if (ab)2 = a?b? in a group G, then ab = ba %3D

A: A group is a set with a binary operation with following axioms satisfied. First the operation must…

Q: 10) Which of the following is not a group? * (Z,*), a* b = a + 2b Va, b E Z O (Z,*), a* b = a +b Va,…

A: Option (1) is correct.

Q: (b) Suppose G is a group, H, K < G, |H|= 30, |K| = 20, and |HN K| = 10. What is |HK|? %3D %3D

A: By theorem, Order of product of two subgroup of finite order       Let the two subgroups be H and K…

Q: In a group G, the law ab-Dac implies b-c is called..

A: Answer : 1 right cancelation law

Q: Show that (ℤ,∗)?ℎ??? r ∗s = (r +s)−(r ∙ s)??? ??? r,s ∈ ℤ is group using variable r, s and t.

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Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')

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Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

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

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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: 2. Show that in a group if x has inverse y and y and a right an inverse r, then y and r are the same…

A: We need to show that in group if x has an inverse y and a right inverse r, then y and r the same…

Q: Consider the subgroup H = (i) of the the group (C\{0},.). LIT TT TT?

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

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Q: Define * on Q by a +b= qb Is Q a group under *? 210

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Q: TASK 2: Prove that if G is a group and a, b = G, then (1) o(a¹ba) = o(b) o(ab) = o(ba)

A: These two results follow by proving G is abelian. We present the detailed solution in the next step.…

Q: Given the group (S = {a,b,c, d},8) %3D a d a b a b d - d The inverse of an element z, z e Sis a a с…

A: The solution is given as

Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

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…

A:

Q: 64

A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…

Q: What is the center of a dihedral group Dn, when n is odd? {H,V} {Ro} {Ro, R180} {Ro, R90, R180,…

A: Solution

Q: 14.Consider the group (A, *) where * is defined by the table below *A B C D ABA D C BA B с D C D C A…

A: