How to know the number of iso/homo morphisms?

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In summary, there are two homomorphisms between S3 and Q8. The first is from order 1 to order 2, and the second is from order 2 to order 3.
  • #1
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Hello everyone. I have been study a little of group theory, i am a little stuck in how to answer question like this:

"How many homomorphism #f: S_{3}\to Q_{8}# are there? $S_{3}$ and $Q_8$ are the permutations group and the quaternion group, respectively."

A homomorphism is a map from A to B such that $\phi(a') \phi(a'') = \phi(a' a '')$, but how to apply this definition to answer the question?

I could construct the multiplicative table for $Q_{8}$, maybe call i j k 1 -i -j -k -1 as 1 2 3 ... 8, but yet have no idea what to do. Maybe seek a "little table" inside this modified table of Q8 that looks like S3?
 
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  • #2
##S_3## is a group of order 6 and ##Q_8## is a group of order 8, right off the bat that should tell you there are no isomorphisms.

As far as homomorphisms, I think that statement alone restricts what kind of images and kernels if can have.
 
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  • #3
These groups are finite, so all elements are of finite order. ##1=\phi(a^n)=(\phi(a))^n## shows, that an element of order ##n## is mapped to an element of order ##k|n##.

Another way is to look at the normal subgroups. The kernel of a homomorphism is a normal subgroup.

All such things have to happen for a homomorphism.
 
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  • #4
Thank you. Okay so let's see...
We have in S3 elements of order 1,2,3 (three are order 2)
In Q8, we have one element order 1, one element order 2 and six elements order 4. This means that we can make a map for the elements of order 1, and 1 map for the elementa of order 2? So the number of homom/ is 2?

So following in this way, and for the sake of clarity, the nunber of homomorphism of another map, which do the opposite, that is, go from Q8 to S3, would be 1 (o1) + 3 (o2) + 3 (o4) = 7?
 
  • #5
No, there are more homomorphisms that are at least potentially available given your analysis. Given an element of order 2 in ##S_3##, how many elements in ##Q_8## can you map it to?
 
  • #6
Office_Shredder said:
No, there are more homomorphisms that are at least potentially available given your analysis. Given an element of order 2 in ##S_3##, how many elements in ##Q_8## can you map it to?
I am not sure for what question was your no, sorry. I will suppose it was to the second question, so i will go on:

S3 has six elements.
S3 ((123),(213),(132),(321),(231),(312))
(123) has order 1
(213),(132),(321) has order 2
(231),(312) has order 3

Q8 has eight elements
1 has order 1
+-i,+-j,+-k has order 4
-1 has order 2

"an element of order n is mapped to an element of order k/n"
Since homomorphism is a group, we can in principle guess this:
I will make an analogy with functions
$\phi(-1) = (213)$,$\phi(-1) = (321)$,$\phi(-1) = (132)$
For each map cited, we can say yet another map, $\phi(1)$, it can be mapped to any elements of S3, so that we have: 3*6 = 18 type of homomorphism in principle.

Now i think we need to check if it is in fact an homomorphism?

There is another way, that i don't know very well how to do too (All books i searched do not have theory of counting homomorphism :S and i am an idiot), that i am supposed to count the kernels. I think we can let this way for later
 
  • #7
Your notation is confusing. You write down the images of ##(123)##, but usually we use another notation:

##(123)## means ##(1\to 2\to 3 \to 1)##. With this more common notation, we have ##S_3=\{(1),(12),(13),(23),(123),(132)\}.##

You have forgotten the element ##1\in Q_8## which can also be the image of an element of order ##2##. You also confused the order: are we talking about homomorphisms ##S_3\longrightarrow Q_8## or ##Q_8\longrightarrow S_3.## This is not a symmetric property, hence direction matters.

Let's stick with the original question ##\phi : S_3\longrightarrow Q_8##.

Thus we can map all three elements of order ##2##, ##(12),(23),(13)## to either ##1## or ##-1##.
This in mind, where do the elements of order three, ##(123),(132)## go to?
 
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  • #8
fresh_42 said:
Your notation is confusing. You write down the images of ##(123)##, but usually we use another notation:

##(123)## means ##(1\to 2\to 3 \to 1)##. With this more common notation, we have ##S_3=\{(1),(12),(13),(23),(123),(132)\}.##

You have forgotten the element ##1\in Q_8## which can also be the image of an element of order ##2##. You also confused the order: are we talking about homomorphisms ##S_3\longrightarrow Q_8## or ##Q_8\longrightarrow S_3.## This is not a symmetric property, hence direction matters.

Let's stick with the original question ##\phi : S_3\longrightarrow Q_8##.

Thus we can map all three elements of order ##2##, ##(12),(23),(13)## to either ##1## or ##-1##.
This in mind, where do the elements of order three, ##(123),(132)## go to?
I realized now that i was misinterpreting your citation in the other comment...
n is mapped to k|n, i was reading n is mapped to k/n...

In s3 we have elements of order 1,2,3.
In q8 we have elements of order 1,2,4.

Elements of order 2 in S3 can be mapped, as you observed, to elements of order 1 and 2 in Q8.
Elements of order 3 in S3 can be mapped to elements of order k|3, namely 1.

##(123),(132)## go to 1.

Ok let me try.

Suppose (12) is mapped to 1, (23) to -1.

##\phi((12)(23)) = \phi(12)\phi(23) = 1*(-1) = -1##
But ##(12)(23) = (123)##
So that ##\phi((123)) = -1##
But it is not allowed...
Maybe we can generalized that (i didn't prove that)
I think we can see for another cases that ##\phi(e_{2}(o2)) = \phi(e_{1}(o2))##
where ##e_{i}(o2)## are elements of order 2 in S3
That is, elements of order 2 in S3 is mapped to equal elements in Q8.

So that there are two maps.
 
  • #9
LCSphysicist said:
I realized now that i was misinterpreting your citation in the other comment...
n is mapped to k|n, i was reading n is mapped to k/n...

In s3 we have elements of order 1,2,3.
In q8 we have elements of order 1,2,4.

Elements of order 2 in S3 can be mapped, as you observed, to elements of order 1 and 2 in Q8.
Elements of order 3 in S3 can be mapped to elements of order k|3, namely 1.

##(123),(132)## go to 1.
Yes.
Ok let me try.

Suppose (12) is mapped to 1, (23) to -1.

##\phi((12)(23)) = \phi(12)\phi(23) = 1*(-1) = -1##
But ##(12)(23) = (123)##
So that ##\phi((123)) = -1##
But it is not allowed...
We already know that there are three elements in the kernel: ##(1),(123),(132),## and as just shown by you, that in case one transposition (2-cycle) is mapped to ##1##, all others have to map onto ##1,## too. This means, as soon as one transposition is in the kernel, then the entire group is, which is the trivial homomorphism. The only question left is: Can we map all three transpositions on ##-1##?

And: Do you know the name of this homomorphism?
Maybe we can generalized that (i didn't prove that)
I think we can see for another cases that ##\phi(e_{2}(o2)) = \phi(e_{1}(o2))##
where ##e_{i}(o2)## are elements of order 2 in S3
That is, elements of order 2 in S3 is mapped to equal elements in Q8.

So that there are two maps.
 
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FAQ: How to know the number of iso/homo morphisms?

1. How do I determine the number of isomorphisms between two structures?

The number of isomorphisms between two structures can be determined by counting the number of possible mappings between the elements of the two structures that preserve their structure and relationships. This can be done by examining the symmetries and transformations of the structures and considering how they can be mapped onto each other.

2. Is there a formula for calculating the number of isomorphisms?

There is no general formula for calculating the number of isomorphisms between two structures. It depends on the specific structures and their symmetries. However, for some simple structures, such as graphs or groups, there are known formulas that can be used to determine the number of isomorphisms.

3. Can the number of isomorphisms change if the structures are modified?

Yes, the number of isomorphisms can change if the structures are modified. Adding or removing elements, changing the relationships between elements, or altering the symmetries of the structures can all affect the number of isomorphisms between them.

4. How does the number of isomorphisms relate to the complexity of a structure?

The number of isomorphisms can be an indication of the complexity of a structure. Generally, the more isomorphisms between two structures, the more similar and interchangeable they are, and the less complex they are. On the other hand, structures with fewer isomorphisms tend to be more unique and complex.

5. Are there any practical applications for knowing the number of isomorphisms?

Yes, knowing the number of isomorphisms between two structures can have practical applications in various fields such as computer science, chemistry, and mathematics. It can help in identifying patterns and similarities between structures, designing efficient algorithms, and predicting the behavior of complex systems.

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