Group homomorphisms between cyclic groups

In summary, a group homomorphism is a function \phi that maps elements from one group (C_4) to another group (C_6) while preserving the group structure. This means that for any two elements a and b in C_4, \phi(a*b) = \phi(a) * \phi(b). The function also maps inverses to inverses. It is important to note that the value of \phi(1) determines the value of \phi at other elements in C_4. Additionally, a general fact about homomorphisms is that the order of \phi(g) must divide the order of g. This can be proven using the definition of a homomorphism.
  • #1
Gott_ist_tot
52
0
Describe al group homomorphisms [tex] \phi [/tex] : [tex] C_4 [/tex] --> [tex] C_6 [/tex]

The book I study from seems to pass over Group Homomorphisms very fast. So I decided to look at Artin's to help and it uses the same definition. So I think I am just not digesting something I should be. I know it's defined as [tex]\phi[/tex] (a*b) = [tex]\phi[/tex] (a) * [tex]\phi[/tex] and that it maps the inverses to the inverses but I just have no idea how to apply these.
 
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  • #2
I forgot a b in the definition of phi(a*b) = phi(a)*phi(b)
 
  • #3
If you specify [itex]\phi(1)[/itex], then what does this say about the value of [itex]\phi[/itex] at the other elements in C_4? Also, a general fact about homomorphisms is that the order of [itex]\phi(g)[/itex] must divide the order of g. Can you prove this? By the way, you can edit posts.
 

1. What is a group homomorphism between cyclic groups?

A group homomorphism between cyclic groups is a function that maps elements from one cyclic group to another while preserving the group operation. In other words, it is a function that maintains the same structure and relationships between elements in both cyclic groups.

2. How can we determine if a group homomorphism between cyclic groups is injective?

A group homomorphism between cyclic groups is injective if and only if the kernel of the homomorphism is trivial, meaning that the only element that maps to the identity element is the identity element itself. In other words, no other elements map to the identity element.

3. Can a group homomorphism between cyclic groups be surjective?

Yes, a group homomorphism between cyclic groups can be surjective. This means that every element in the target cyclic group has at least one preimage in the source cyclic group. In other words, the function covers the entire target group.

4. How does the order of the cyclic groups affect the group homomorphism?

The order of the cyclic groups can affect the group homomorphism in a few ways. If the order of the target cyclic group is smaller than the order of the source cyclic group, then the homomorphism cannot be surjective. Also, if the orders of both cyclic groups are relatively prime, then the homomorphism must be injective.

5. What is the significance of group homomorphisms between cyclic groups?

Group homomorphisms between cyclic groups are significant because they allow us to better understand the relationship between different cyclic groups. They also play an important role in abstract algebra, as they help us to study the properties and structures of groups in a more general setting.

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