# Thread: Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

1. ## Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.

2. Originally Posted by rainyice
Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.

Too many questions , too little work shown: what've you done to solve these questions? Where are you stuck?

Tonio

3. Originally Posted by rainyice
Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.
Hint : map a generator to a generator, and extend the map to a homomorphism in the only possible way; then show that it's an isomorphism.

4. ## Working on the same problem

Originally Posted by rainyice
Prove that any cyclic group of finite order n is isomorphic to Zn under addition.

Should I draw two tables to indicate that? Like ... the first one is cyclic group table G and the second one is the Zn table H, then I linked it G -> f -> H where f is the transformation.
I don't think drawing tables qualifies as a proof (I'm working on the same problem). Also, you weren't told what operation "G" was under, so you wouldn't be able to make a table for cyclic group G.

I believe we have to show that some generator <x> of G maps into some generator <a> of Z. To show isomorphism, we have to satisfy two conditions. To show
φ (some function) is a one-to-one correspondence from G to Z, we need to show that every element in G maps into every element in Z. For this one, I'm assuming since both groups are of the same order (we stated that cyclic group G is of order n and it is given that Z is of order n), there is automatically a one-to-one correspondence between the two groups (I'm not sure though).

I'm still working on showing the second condition of isomorphism.

5. Originally Posted by MissMousey
I don't think drawing tables qualifies as a proof (I'm working on the same problem). Also, you weren't told what operation "G" was under, so you wouldn't be able to make a table for cyclic group G.

I believe we have to show that some generator <x> of G maps into some generator <a> of Z. To show isomorphism, we have to satisfy two conditions. To show
φ (some function) is a one-to-one correspondence from G to Z, we need to show that every element in G maps into every element in Z. For this one, I'm assuming since both groups are of the same order (we stated that cyclic group G is of order n and it is given that Z is of order n), there is automatically a one-to-one correspondence between the two groups (I'm not sure though).

I'm still working on showing the second condition of isomorphism.

Some people call me the gangster of love.

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# every cyclic group is isomorphic to zn

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