Question

every homomorphic image of a cyclic group is cyclic​

Answer 1

Answer:

Step-by-step explanation:

Our aim is to prove that every homomorphic image of a cyclic group is cyclic​.

Lets consider f: G → H to be an homomorfism.

Now, suppose that G is generated by <g>.

Hence, if h is in the range of f, thus, we can say that there exists one element in G which satisfies f(a) = h.

We know that a is an element of G, hence it exists an n natural number such that a = g^n.

From this we can see that h = f(a) = f(g^n) = f(g)^n .

Hence, we can express any image of a cyclic group as a power of the image of a generator of the cyclic group.

Then we conclude that every homomorphic image of a cyclic group is cyclic.

I hope this helps you!! Keep up the good studies!!