# Isomorphic groups

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 $(\langle i\rangle,\cdot)$?

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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?

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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##?

Mentor

## 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.

gruba said:

## 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

Mentor
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.
gruba said:
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 $f$?

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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.

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$f(x)=e^x$ is one isomorphism.
Not between the given groups.

gruba
Not between the given groups.
$f(x)=e^{2\pi x i}$?

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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?

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