Let G and G1 be two groups such that G1 is a homomorphic image of G.
Prove that if G is a commutative group,then so is G1.
Here it is given that G1 is a hmomorphic image of G.
So,there exists a function f: G to G1.
Suppose G is commutative.
Let us take arbitrary two elements a,b belongs to G1 = f(G).
Then there exists two elements x and y belongs to G such that f(x) = a and f(y) = b.
Now, ab = f(x)f(y) = f(xy) (Definition of homomorphism) = f(yx) (Commutativity of G) = f(y)f(x) = ba.
Hence G1 is a commutative group. (Proved)