Help Proving Isomorphism of a group

In summary, the conversation discusses proving a cyclic group is isomorphic to a ring under addition and the strategy to take. The conditions of a group being isomorphic are mentioned, as well as the concept of homomorphic with onto and 1 to 1. The conversation then delves into defining an explicit isomorphism and defining $f(g)=g^n$ (mod n). The conversation also touches on the theory behind why this proof works. Finally, the conversation clarifies the definition of $f$ and the role of the generator $g$ in proving the isomorphism.
  • #1
Bruce Wayne1
15
0
Hi!

I'm trying to prove a cyclic group is isomorphic to ring under addition. What the strategy I would take? How would I get it started?

Here's what I know so far:

I need to meet 3 conditions-- 1 to 1, onto, and the operation is preserved. I also know that isomorphic means that the group is homomorphic with onto and 1 to 1.
 
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  • #2
Bruce Wayne said:
I'm trying to prove a cyclic group is isomorphic to ring under addition. What the strategy I would take? How would I get it started?
You need to come up with an explicit definition of an alleged isomorphism.
 
  • #3
are you tring to prove a cyclic group G of order n is isomorphic to Z_n ={0,...,n-1}?Define $f(g)=g^n$ (mod n)
 
  • #4
Poirot said:
are you tring to prove a cyclic group G of order n is isomorphic to Z_n ={0,...,n-1}?Define $f(g)=g^n$ (mod n)
I may be missing something. Is $f:G\to\mathbb{Z}_n$? Then what is $g^n\pmod{n}$?
 
  • #5
Poirot said:
are you tring to prove a cyclic group G of order n is isomorphic to Z_n ={0,...,n-1}?Define $f(g)=g^n$ (mod n)

Yes, exactly. I want to show that a cyclic group G of order n is isomorphic to Zn.

I understand the concepts, and I know how to prove it for a relatively small n, but I haven't been able to find a completed proof online. I'd like to read it, and ask a few questions on the theory of why it works (should any arise).
 
  • #6
Since $G$ is cyclic of order $n$, there exists a $g\in G$ such that every $x\in G$ can be represented as $g^k$ for some $0\le k<n$. Therefore, we can define an $f:G\to\mathbb{Z}_n$ as follows: $f(g^k)=k$. It rests to show that $f$ is one-to-one, onto and a homomorphism.
 
  • #7
Evgeny.Makarov said:
I may be missing something. Is $f:G\to\mathbb{Z}_n$? Then what is $g^n\pmod{n}$?

yes sorry I mean f(g^k)=k (mod n), where g is the generator of G
 

FAQ: Help Proving Isomorphism of a group

What is isomorphism in the context of a group?

In mathematics, isomorphism refers to a type of mapping between two mathematical structures that preserves their underlying structure and operations. In the context of groups, isomorphism means that two groups have the same structure and operations, but may differ in their elements and notation.

Why is it important to prove isomorphism of a group?

Proving isomorphism of a group is important because it allows us to show that two seemingly different groups are actually equivalent in terms of their structure and operations. This can help us make connections between different groups and simplify complex problems by using the properties of isomorphic groups.

What are the steps to prove isomorphism of a group?

The general steps to prove isomorphism of a group are as follows:

  1. Define a mapping between the two groups that you want to prove are isomorphic.
  2. Show that the mapping is one-to-one, meaning that each element in one group maps to a unique element in the other group.
  3. Show that the mapping is onto, meaning that every element in the second group has a corresponding element in the first group.
  4. Verify that the mapping preserves the group operations, meaning that the result of the operation in one group is the same as the result of the operation in the other group after mapping.
  5. Conclude that the two groups are isomorphic.

What are some common techniques used to prove isomorphism of a group?

Some common techniques used to prove isomorphism of a group include:

  • Constructing a bijection between the two groups.
  • Using the properties of group operations to show that the mapping preserves the structure.
  • Using the Cayley table of the groups to show that they are identical after mapping.
  • Using subgroup and quotient group properties to prove isomorphism.

Are there any tools or software available to help with proving isomorphism of a group?

Yes, there are several tools and software available to assist with proving isomorphism of a group. Some popular options include GAP, MAGMA, and Maple. These programs have built-in functions and algorithms for checking isomorphism and can also help with constructing bijections between groups.

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