Let G and H be groups, and let N be a normal subgroup of G. Decide whether each of the following two statements is true or false. Give a proof or a counterexample in each case. (a) The group G × H is abelian if and only if G and H are both abelian. (b) The group G/N is abelian if and only if both G and N are abelian.

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A: The given statement is Any subgroup of F2 (the free group upon 2 generators) is not abelian.

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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: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

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

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Q: Consider the following statements B. Every cyclic group is abelian C. Every abelian group is cyclic…

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Q: Prove that Sn is not abelian for all n > 3.

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

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

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

A: The objective is to show that the given group is abelian. Given group is: a,b,c|b4=a,c-1=b

Q: Exercise 8.1. Let G₁, G2, ..., Gn be a finite collection of groups. Prove that and only if each G;…

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Q: 9. If : G→ H is a group homomorphism and G is abelian, prove that (G) is also abelian.

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