Abelian group isomorphism help

In summary, the conversation discusses the concept of a homomorphism, phi, which is a map from an abelian group, G, to itself, sending each element to its nth power. It is shown that the kernel of phi is the set of elements in G whose order divides n. Additionally, it is mentioned that phi is an isomorphism if n is relatively prime to the order of G. The conversation ends with a suggestion to start with definitions when unsure of how to approach a problem.
  • #1
b0mb0nika
37
0
let G be an abelian group, and n positive integer
phi is a map frm G to G sending x->x^n
phi is a homomorphism

show that
a.)ker phi={g from G, |g| divides n}
b.) phi is an isomorphism if n is relatively primes to |G|

i have no clue how to even start the prob...:-(
 
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  • #2
If you're using |g| to mean the order of g in G, then you've made a typo in part (a).
 
  • #3
yes i did..
|g| divides n
sorry about that :)
 
  • #4
Well, when you have no clue where to begin, the definitions are often a very good place to start.
 
  • #5
A. Let [itex]x\in G[/itex]. Then [itex] x^n=e \Leftrightarrow O(x)|n \Rightarrow Ker(\phi)=\{g\in G| O(g)|n\} [/itex]

B. I have to go, I might come back later if it isn't solved by then.
 

What is an Abelian group?

An Abelian group, also known as a commutative group, is a mathematical structure consisting of a set of elements and an operation that is commutative, meaning the order in which the operation is performed does not affect the result. It is named after mathematician Niels Henrik Abel.

What is an isomorphism?

An isomorphism is a mathematical concept that describes a one-to-one mapping between two structures. In the context of Abelian groups, an isomorphism is a function that preserves the group operation, meaning the operation performed on two elements in one group will result in the same element if performed on their mapped counterparts in the other group.

Why is Abelian group isomorphism important?

Abelian group isomorphism is important because it allows us to study and compare different groups by finding common structures and properties. It also helps us simplify complex group structures by identifying isomorphic groups that behave in the same way.

How do you prove two Abelian groups are isomorphic?

To prove two Abelian groups are isomorphic, you must show that there exists a function between them that is both one-to-one and preserves the group operation. This can be done by explicitly constructing the function or by showing that the groups have the same structure and can be mapped onto each other.

What are some applications of Abelian group isomorphism?

Abelian group isomorphism has applications in various fields of mathematics, including number theory, group theory, and algebraic geometry. It is also used in computer science for data compression and error-correction coding, as well as in physics for studying symmetries and conservation laws.

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