Question

let N be a normal subgroup of G . show that if G is abelian , then G/N is also abelian​

Answer 1

Answer:

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Step-by-step explanation:

A subgroup is normal if the left and right cosets coincide. That is if x+N={x+a|a∈N}=N+x={a+x|a∈N}x+N={x+a|a∈N}=N+x={a+x|a∈N}. In the case of abelian groups this is obviously true so all subgroups of an abelian group are normal. Now take any two elements in G/NG/N say x+Nx+N and y+Ny+N then we must show that (x+N)+(y+N)=(x+y)+N(x+N)+(y+N)=(x+y)+N is the same as (y+N)+(x+N)=(y+x)+N(y+N)+(x+N)=(y+x)+N. I leave this as an exercise (hint G is abelian).