Prove Isomorphic Groups: (\mathbb Z_4,_{+4}) and (\langle i\rangle, \cdot)

[gruba]

Homework Statement

Show that the group $(\mathbb Z_4,_{+4})$ is isomorphic to $(\langle i\rangle,\cdot)$?

Homework Equations

-Group isomorphism

The Attempt at a Solution

Let $\mathbb Z_4=\{0,1,2,3\}$.
$(\mathbb Z_4,_{+4})$ can be represented using Cayley's table:
$$\begin{array}{c|lcr} {_{+4}} & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 0 \\ 2 & 2 & 3 & 0 & 1 \\ 3 & 3 & 0 & 1 & 2 \\ \end{array}$$

What is the set $\langle i\rangle$?
How to define $(\langle i\rangle,\cdot)$?

 

[micromass]

The subgroup of $(\mathbb{C},\cdot)$ generated by $i$.

 

[gruba]

The subgroup of $(\mathbb{C},\cdot)$ generated by $i$.

What should be the order of that subgroup, and how to represent it using Cayley's table?

 

[Orodruin]

What should be the order of that subgroup, and how to represent it using Cayley's table?

Why don't you try figuring it out? What is $i^2$?

 

[Mark44]

Homework Statement

Show that the group $(\mathbb Z_4,_{+4})$ is isomorphic to $(\langle i\rangle,\cdot)$?

Should this be $(\mathbb{Z_4}, +)$?
A group is defined by a set of elements of the group, together with an operation.

Homework Equations

-Group isomorphism

The Attempt at a Solution

Let $\mathbb Z_4=\{0,1,2,3\}$.
$(\mathbb Z_4,_{+4})$ can be represented using Cayley's table:
$$\begin{array}{c|lcr} {_{+4}} & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 0 \\ 2 & 2 & 3 & 0 & 1 \\ 3 & 3 & 0 & 1 & 2 \\ \end{array}$$

What is the set $\langle i\rangle$?
How to define $(\langle i\rangle,\cdot)$?

 

[gruba]

Could someone explain this problem (using Cayley's tables - easier)? How to form Cayley's table for the group $(\langle i\rangle,\cdot)$?

One method to show the groups are isomorphic is to create Cayley's tables and compare them (that is only useful for small groups).
I don't understand the method which requires finding the function (isomorphism) between these groups

 

[Mark44]

Could someone explain this problem (using Cayley's tables - easier)? How to form Cayley's table for the group $(\langle i\rangle,\cdot)$?

Why don't you try what Orodruin suggested -- find i², i³, and so on. This is not a hard problem.

One method to show the groups are isomorphic is to create Cayley's tables and compare them (that is only useful for small groups).
I don't understand the method which requires finding the function (isomorphism) between these groups

 

[gruba]

Why don't you try what Orodruin suggested -- find i², i³, and so on. This is not a hard problem.

Let $f:\mathbb Z_4\rightarrow \langle i\rangle=\{i^0,i^1,i^2,i^3\}=\{1,i,-1,-i\}$ where $f$ is an isomorphism.
From here, how to explicitly define a function $f$?

 

[Orodruin]

From here, how to explicitly define a function fff?

What do you think? There are only four possibilities of defining a homomorphism (it is fully defined by specifying how f acts on the group generator). Two of them give isomorphisms!

 

[gruba]

What do you think? There are only four possibilities of defining a homomorphism (it is fully defined by specifying how f acts on the group generator). Two of them give isomorphisms!

$f(x)=e^x$ is one isomorphism.

 

[Orodruin]

$f(x)=e^x$ is one isomorphism.

Not between the given groups.

 

[gruba]

Not between the given groups.

$f(x)=e^{2\pi x i}$?

 

[micromass]

If I define $f(0) = 1$ and if I say $f$ is a homomorphism, can you figure out $f(1)$, $f(2)$ and $f(3)$? That is, can you describe $f$ completely??

 

[gruba]

If I define $f(0) = 1$ and if I say $f$ is a homomorphism, can you figure out $f(1)$, $f(2)$ and $f(3)$? That is, can you describe $f$ completely??

$f(0)=1,f(1)=i,f(2)=-1,f(3)=-i$.

Using Lagrange interpolation polynomial on points $(0,1),(1,i),(2,-1),(3,-i)$ gives
$f(x)=-\frac{(x-1)(x-2)(x-3)}{6}+i\frac{x(x-2)(x-3)}{2}+\frac{x(x-1)(x-3)}{2}-i\frac{x(x-1)(x-2)}{6}$.

But $f(x)$ is not one to one.

What is the actual method for describing an isomorphism, without taking a guess?

 

[LCKurtz]

@gruba I'm guessing that the lack of further replies is caused by your last reply. It appears to me that you need more help than can be provided under the rules of this forum. My suggestion to is that you need to schedule a personal meeting with your teacher to clear up your misunderstandings on this topic.

 

[Orodruin]

But f(x) is not one to one.

It is one-to-one on the relevant sets. You have specified f(x) for all elements of $\mathbb Z_4$, there is absolutely no need to express it in terms of a polynomial (why would you think there was?).