Mapping generator to generator in cyclic groups.

Thread starter jmjlt88
Start date Jun 22, 2012
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Cyclic Generator Groups Mapping

In summary, a generator in cyclic groups is an element that can generate all other elements in the group by repeated multiplication. It is also known as a primitive element. To map one generator to another in cyclic groups, a group homomorphism must be found that preserves the generator. This means that the mapping must preserve the group structure and the generator must still generate all other elements in the group after the mapping is applied. A group homomorphism is a function that preserves the group structure, and in the context of mapping generators in cyclic groups, it must preserve the generator of the group. Mapping generators in cyclic groups is important for understanding relationships between different generators, identifying patterns and symmetries, and solving problems and proving theorems. However

Jun 22, 2012

jmjlt88

Attached is my attempt at a proof. Please critque! :shy: Thank you!

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Jun 22, 2012

jmjlt88

Note: Equation (1) in my proof is something that I proved in a prior exercise.

I did it by Mathematical Induction. I don't have it typed nicely, but I am pretty confident it worked.

Jun 25, 2012

christoff

Looks good! I don't see any issues.

1. What is a generator in cyclic groups?

A generator in cyclic groups is an element that can generate all other elements in the group by repeated multiplication. It is also known as a primitive element.

2. How do you map one generator to another in cyclic groups?

To map one generator to another in cyclic groups, you need to find a group homomorphism that preserves the generator. This means that the mapping must preserve the group structure and the generator must still generate all other elements in the group after the mapping is applied.

3. What is a group homomorphism?

A group homomorphism is a function that preserves the group structure, meaning that the result of applying the function to two elements in the group is the same as applying the group operation to the two elements first and then applying the function to the result. In the context of mapping generators in cyclic groups, the group homomorphism must preserve the generator of the group.

4. Why is mapping generators in cyclic groups important?

Mapping generators in cyclic groups is important because it allows us to understand the relationship between different generators in the group. It also helps us to identify patterns and symmetries within the group, which can be useful in solving problems and proving theorems.

5. Are all generators in cyclic groups mappable to each other?

No, not all generators in cyclic groups are mappable to each other. A mapping can only be established between two generators if there exists a group homomorphism that preserves the generator. In some cases, such a mapping may not exist, and in other cases, there may be multiple possible mappings.