If G is abelian then G/Z(G) is cyclic. *True *False

Q: For any two elements a and b of a group G is abelian iff (ab)² = a²b². , show that G

A: For any two elements a and b of a group GClaim : G is abelian iff (ab)2 =a2b2 .

Q: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: If G is non-Abelian, show that Aut(G) is not cyclic.

A: Given: G is non-abelian. To Show: Aut(G) is not cyclic.

Q: True or False with proof "Any free abelian group is a free group."

A: Given statement is "Any free abelian group is a free group."

Q: Let G = .  Show that the permutations (23) and (13) satisfy the defining relations of G.  Explain…

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Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e a^4 =…

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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: Question 3: Leb 3:6>G' be an isomorphism . Prove the follawing assertions : la) IGl =1G| (b) lal =I…

A: We have to solve given problem:

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

A: We need to prove that a group G is abelian if and only if (ab)-1=a-1b-1 , for all a,b in G. The…

Q: lternating group An is abelian if and only if n <=3.

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Q: /rite Z 51 X Z260 as a product of cyclic groups, both in long form, and in Smith normal form.

A: This is abstract algebra problem of classification of finitely generated abelian groups. Please find…

Q: Find all abelian groups upto isomorphism  of order 4 and 6.

A: We have to find the all abelian groups upto isomorphism of order 4 and 6.

Q: If N and K are submodules of M. (i) The kernel of the natural homomorphism N--> M/K is (N+K)/K. (ii)…

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Q: Suppose that f: G → G such that f(x) and only if = axa. Then f is a group homomorphism if a = e a^4…

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Q: Determine the order of (Z ⨁ Z)/&lt;(2, 2)&gt;. Is the group cyclic?

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

Q: Show that Every quotient group of abelian grap Gabelian but converse is not tow. I u

A: We have to show that every quotient group of an abelian group G is abelian but converse is not true.

Q: Suppose that f: G G such that f(x) and only if = axa. Then f is a group homomorphism if -> a = e a^3…

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Q: Give an example of two non-homomorphic finite abelian groups that have the same number of elements…

A: We need to find an example of two non holomorphic finite abelian groups which have the same number…

Q: Given two examples of finite abelian groups

A: Require examples of finite abelian groups.

Q: Show that G1XG2 is abelian if and only if both G1 and G2 are abelian.

A: We have to show that is abelian if and only if both are abelian.

Q: Find the group homomorphism between (Z, +) and (R- (0},.)

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Q: Every quotient group of an abelian group G  is abelian but converse is not true.

A: Every quotient group of an abelian group G is abelian but converse is not true.

Q: Prove that An is not abelian for all n > 4.

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Q: b) Prove that if G is abelian and H ¶G, then G/H is abelian.

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Q: Let (G1, •) and (G2 ,*) be two groups and o: G1 G2 be an isomorphism. Then O G2 might be abelian…

A: Since phi : G1 --&gt; G2 is an isomorphism hence G1 and G2 have same properties .

Q: Prove that if G/Z(G) is cyclic, then G is abelian.

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