Group theory

Question

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.

Answer 1

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)