# Homomorphism of a cyclic subgroup is a cyclic subgroup ?

• Leb
In summary, the conversation discusses the proof of the statement α(<x>) = <α(x)>, given that α is a homomorphism from group G to group H and x is an element of G. The proof involves showing that for any r in the set of integers, α(x^r) is equal to α(x)^r. This is achieved by using the definition of a homomorphism and the binary operations in groups G and H.
Leb

## Homework Statement

Let $\alpha:G \rightarrow H$ be a homomorphism and let x$\in$G

Prove $\alpha(<x>) =<\alpha(x)>$

## Homework Equations

α(<x>) = α({x^{r}: r ∈ Z}) = {α(x^{r}) : r ∈ Z} = {α(x)^{r}: r ∈ Z} = <α(x)>.

I do not understand how can we take out the 'r' out of a(x^{r}) to get (a(x))^{r} ? What would a(x)^2 say mean ? a(x) (some binary operation defined in H) a(x) or would it be (a°a)(x) i.e. a(a(x)) ?

Hi Leb!

(try using the X2 button just above the Reply box )
Leb said:
I do not understand how can we take out the 'r' out of a(x^{r}) to get (a(x))^{r} ? What would a(x)^2 say mean ? a(x) (some binary operation defined in H) a(x) or would it be (a°a)(x) i.e. a(a(x)) ?

No, that would be ar(x).

This is (a(x))r, the product of r elements of group H.

Thank you for your reply tiny-tim !
However, I am still not sure what will happen...

(a(x))r = a(x) * a(x) *...*a(x) r-times ? But what is * then ? Is it defined in H or in G ?

Leb said:
I do not understand how can we take out the 'r' out of a(x^{r}) to get (a(x))^{r} ?
Well, you have that $a$ is homomorphism, so $a\bigl(x^2\bigr)=a(x\star x)=a(x)*a(x)=\bigl(a(x)\bigr)^2$, same for $a(x^{r})$, where $\star$ is binary operation in $G$ and $*$ in $H$.

Leb said:
… I am still not sure what will happen...

(a(x))r = a(x) * a(x) *...*a(x) r-times ? But what is * then ? Is it defined in H or in G ?

in H

x is in G, a(x) is in H …

so x can only undergo G's operations,

and a(x) can only undergo H's operations

Ah, OK, now it is clear. Thanks guys !

## 1. What is a homomorphism of a cyclic subgroup?

A homomorphism of a cyclic subgroup is a function that preserves the group structure of a cyclic subgroup. This means that the function maps the elements of the subgroup to another group in a way that respects the group operation.

## 2. What is a cyclic subgroup?

A cyclic subgroup is a subgroup of a group that is generated by a single element. This means that the subgroup only contains elements that can be obtained by repeatedly applying the group operation to the generator element.

## 3. How do you determine if a homomorphism of a cyclic subgroup is a cyclic subgroup?

To determine if a homomorphism of a cyclic subgroup is a cyclic subgroup, you can check if the image of the generator element under the function also generates the image group. If it does, then the homomorphism is a cyclic subgroup.

## 4. What is the importance of a homomorphism of a cyclic subgroup being a cyclic subgroup?

A homomorphism of a cyclic subgroup being a cyclic subgroup is important because it allows us to easily understand the structure and behavior of the subgroup. This can help with proving theorems and solving problems related to the subgroup.

## 5. Can a homomorphism of a cyclic subgroup be a cyclic subgroup if the subgroup is not cyclic?

No, a homomorphism of a cyclic subgroup can only be a cyclic subgroup if the subgroup itself is cyclic. If the subgroup is not cyclic, then the homomorphism may not preserve the group structure and therefore cannot be a cyclic subgroup.

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