24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are in Z(G).

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: 9. Prove that a group of order 3 must be cyclic.

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Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: Prove O3 is not a group

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Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

A: To Determine: prove that H,. is a subgroup of a group of non zero complex number under…

Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: Let G be a group with the property that for any x, y, z in the group,xy = zx implies y = z. Prove…

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

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Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

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

A:

Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: prove that any group R=3 must beperiedio

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Q: Z, x is not a group.

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Q: 3. Define Lie group.

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Q: (3) Show that 2Z is isomorphic to Z. Conclude that a group can be isomorphic to one of its proper…

A: (2ℤ , +) is  isomorphic to (ℤ , +) .              Define f :(ℤ , +) →(2ℤ , +) by…

Q: Prove or give counterexample. For any group G, Z(G) ≤ [G, G].

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Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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

A:

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

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

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Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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Q: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

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Q: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

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: not

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: determine whether the binary  operation * defined by a*b=ab gives group structure of Z. if it is not…

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Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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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 necessa: Abelian Of…

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Q: Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

A: 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3…

Q: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

A:

Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

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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: 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: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…

Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution