Showing that groups are isomorphic

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In summary, if one wants to show that two groups are isomorphic, it is sufficient to find a single isomorphism between them. For example, if G is an infinite cyclic group with generator g, it can be shown that G is isomorphic to Z with the function f(g) = ord(g). This function is both bijective and a homomorphism.
  • #1
gottfried
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If one wants to show that two groups are isomorphic is simply finding a single isomorphism between them sufficient?

For example.

If G is an infinite cyclic group with generator g show that G is isomorphic to [itex]Z[/itex].

So suppose f(g)=ord(g)

then f is bijective and a homomorphism I believe?
 
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  • #2
gottfried said:
If one wants to show that two groups are isomorphic is simply finding a single isomorphism between them sufficient?

Yes.

For example.

If G is an infinite cyclic group with generator g show that G is isomorphic to [itex]Z[/itex].

So suppose f(g)=ord(g)

then f is bijective and a homomorphism I believe?

Yes, this is true. However, you might want to give a bit of explanation on why it is bijective and a homomorphism.
 
  • #3
Cool thanks for clearing it up.
 

Related to Showing that groups are isomorphic

1. How can I prove that two groups are isomorphic?

There are a few ways to prove that two groups are isomorphic. One method is to find a bijective function (a function that is both one-to-one and onto) between the two groups that preserves the group operation. Another method is to show that both groups have the same structure, meaning they have the same number of elements and the same number of elements of each order.

2. What is the significance of proving two groups are isomorphic?

Proving that two groups are isomorphic means that they have the same algebraic structure, even if their elements and operations may look different. This can help us understand the relationship between different groups and makes it easier to transfer knowledge and theorems from one group to another.

3. Can two groups be isomorphic but have different group presentations?

Yes, two groups can be isomorphic but have different group presentations. Group presentation refers to the way a group is defined by its generators and relations. Isomorphism only looks at the structure of the group, so as long as the two groups have the same structure, they can be isomorphic even if their group presentations are different.

4. Is every group isomorphic to itself?

Yes, every group is isomorphic to itself. This is because the identity function, which maps each element to itself, is always a bijective function that preserves the group operation. Therefore, every group is isomorphic to itself with the identity function as the isomorphism.

5. Can two groups be isomorphic if they have different orders?

No, two groups cannot be isomorphic if they have different orders. The order of a group refers to the number of elements in the group, and isomorphism preserves the number of elements. Therefore, if two groups have different orders, they cannot have the same structure and cannot be isomorphic.

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