1. Show that the group (a, b, c|bª = a, c¬1 = b) is abelian. %3D

Q: Label the following statement as either true or false.   The alternating group A4 on 4 elements is…

A: We have to state whether the given statement is true or false : The given statement is : The…

Q: (c) Prove that if G is a (not necessarily abelian) group, a, b e G, and a² = b² = (ab)² = e, then ab…

A: Use property of group and solve it.

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: Show that any group of order 3 is abelian.

A: The solution is given as

Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

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: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

A:

Q: 3. Let n eN be given. Is the set U = {A: det A = ±1} C Matnxn(R) a group under matrix multipli- %3D…

A: By using properties of group we solve the question no. 3 as follows :

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

Q: One of the following is True O If aH = bH, then Ha = Hb. O Only subgroups of finite groups can have…

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

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Q: be the operation on Z defined by a*b = a+b for all a,beZ. Justify the following questions. 4 Let (1)…

A: Here * be the operation on ℤ defined by a*b=a+b4 for all a, b∈ℤ. We have to justify the followings:…

Q: 3. You have already proved that GL(2, R) = {[ª la, b, c, d e R and ad – bc ± 0} forms a group under…

A: Note: There are two questions and I will answer the first question. So, please send the other…

Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: QUESTION 7 Show that the special linear group, SL(2, R) is non -Abelian.

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Q: Let G be a group. V a,b,c d and x in G, if axb=cxd then ab=cd then G is necessarily: * O Abelian O…

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Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: Let G = (Z,, +6) is an Abelian group then how many self - invertible elements in G? (A) 1 (B) 2 (C)…

A: To solve this problem, we use the defination of group.

Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: 6. ! Write down the relation matrix of the abelian group G specified as follows. G - {x, y, z, w |…

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Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

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

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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: Exercise 2.5.6 State whether each of the following statements is true or false and give a brief…

A: (a) False Let G be a group and a∈G then if a is generator of G then a-1 is also generator of G.…

Q: Consider the group G = {x € R]x # 1} under the binary operation : *• y = xy – x-y +2 The identity…

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Q: If (G, * ) is a group with a a for all a in G then G is abelian

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Q: 52

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Q: Let G = (Z6, +6) is an Abelian group then how many self - invertible elements in G? 1.a O 4 .b O 3.c…

A: Let G = (Z6 , +6) is an abelian group. We know Z6 = {0,1,2,3,4,5} An element is said to be self…

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: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: 2) Prove that Zm × Zn is a cyclic group if and only if gcd(m, n) cyclic group Z; x Z4. = 1. Find all…

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Q: Show that the set S = (1, i, - 1, -0 is an abelian group with respect to the multiplication

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Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: 1- Let (C\{0},.) be a group of non-zero complex number and let H = {a + ib, a² + b² = 1} then (H,.)…

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Q: Let G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

A: First, consider that the group is abelian. So here first compute (ab)-1 for a and b belongs to G,…

Q: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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Q: Prove that is (ab)-1 = a-1b-1 for all a,b in group G, then G is abelian.

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Q: In D4, the centralizer of the group at H is equal to?

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Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D %3D where a = (1 2 3 4)…

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Q: Let G be a group. V a,b,c d and x in G, if axb=cxd then ab=cd then G is necessarily: O Abelian O Of…

A: Solution:Given    G be a group∀a,b,c,d and x in G

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: 64

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

Q: Exercise 5.4.30. (a) Show that the nonzero elements of Zz is a group under o. (b) Can you find an n…

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Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

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Q: Let G be a group. V a, b, c d and x in G, if axb = cxd then ab = cd then G is necessarily:…

A: The answer is given as follows :

Q: Prove that the symmetric group (S₂, 0) is abelian.

A: