Proofs Algebraic

Question 0

Prove. In an isomorphism of groups. Then the unique inverse map ψ^-1 such that ψ^-1oψ = e of G and ψοψ^-1= e of H

Answer 0

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Question 1

Prove C((12…n)) =

Answer 1

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Question 2

Prove for any homomorphism G to H. The kernel of ψ is a normal subgroup of G

Answer 2

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Question 3

Prove for any group G, the centre is a normal subgroup of G

Answer 3

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Question 4

Prove (xN)(yN) = (xy)N is a binary operation on set G/N

Answer 4

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Question 5

Prove. N is normal subgroup of G. The set of left cosets G/N is a group with respect to the binary operation (xN)(yN) = (xy)N

Answer 5

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Question 6

Prove any subgroup of index 2 in a group is normal

Answer 6

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Question 7

Prove that the factor group G/Z(G) is not cyclic, if G is not abelian

Answer 7

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