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

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: Prove that the fundamental group is abelian if and only if each homomorphism γ∗ as above only…

A: Let us assume that π1(X), the fundamental group is abelian. Let us consider a loop α with…

Q: Suppose that G is an Abelian group of order 35 and every element of G satisfies the equation x35 =…

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Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

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: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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Q: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

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Q: The group GLQ,R) abelian group is an

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Q: Exercise: Show that (Z5, +5) is an abelian group.

A: The element of Z5={0,1,2,3,4}

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

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Q: Show that G ⨁ H is Abelian if and only if G and H are Abelian.State the general case

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

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

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: Let G={-1,0,1}. Verify whether G forms an Abelian group under addition.

A: G  is a group under '+' if  (i) a , b E G  -----> a + b E G     (ii) a E G called the identity…

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

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Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

Q: Show that any group of order 4 or less is abelian

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Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e  also x6=e therefore…

Q: 6. Embed the group Qs into the SU(2).

A: Given: Q0=e,i,j,k  e-2=e, i2=j2=k2=ijk=e, Where, e is the identity element and e commutes with the…

Q: 64. Express Ug(72)and U4(300)as an external direct product of cyclic groups of the form Zp

A: see my attachments

Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: Prove that a group G is abelian if and only if (ab)-1 = a¬b¬1 va,bEG

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Q: Which abelian somorphic to groups subyraups of Sc. Explin. are

A: Writing a permutation σ∈Sn as a product of n disjoint circles. i.e σ=τ1,τ2,τ3,…τk The order of σ is…

Q: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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

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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: (G, .> be a group such that a.a = e for all a E G. Show that G is an abelian grou 2. Let

A: We have to solve given problem:

Q: Using the Theorem of Lagrange, prove that a group G of order 9 is abelian

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Q: Prove that (ab)2 = a²b² for all a, b in a group G if and only if G is Abelian.

A: Let G be set and "·" be a binary operation then G,· is a group if  Clouser property. That is, if…

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: Example: Show that (Z,+) is a semi-group with identity

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Q: Prove that An even permutation is group w.y.t compostin Compostin function.

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Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

Q: Determine the galois group

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Q: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

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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: 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: 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

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Q: If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in…

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

A: See the attachment.

Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: 8. Show that (Z,,×s) is a monoid. Is (Z.,×6) an abelian group? Justify your answer

A: Note: since you have posted multiple questions . As per our guidelines we are supposed to solve one…

Q: Prove that a group is abelian if and only if (ab)-l = a-lb-1 va,bEG Attach File Browse Local Files

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Q: OLet a and b be elements of a group G. Prove that G is abelian if and only if (ab)- = a¯\b-!.

A: Prove that G is abelian if and only if (ab)-1=a-1b-1. For all a and b be elements of a group G.

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