Group Theory

Question

Let G and G1 be two groups such that G₁ is a homomorphic image of G.

If G is a cyclic group then so is G₁.

Answer 1

Here it is given that G₁ is a homomorphic image of G.

So, there exist a function f: G to G₁.

Suppose G is a cyclic group.

Let G = <a> (where a is the generator of G).

Hence, f(a) belongs to G₁.

Let, f(a) = b.

Now, we will prove that G₁ = <b> (That is b is the generator of G₁)

Assume an arbitrary element u belongs to G₁.

So, there exist y belongs to G such that f(y) = u.

Since G is cyclic and a is the generator of G we have y = aⁿ

Then u = f(y) = f(aⁿ) = f(a)ⁿ (Definition of homorphism of groups) = bⁿ

Hence, G₁ = <b> and G₁ is a cyclic group (proved)