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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.

asked in Mathematical Logic by (475 points) | 35 views

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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)

answered by Loyal (9.5k points)
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