Show that this is a homomorphism

  • Thread starter ArcanaNoir
  • Start date
  • #1
768
4

Homework Statement


Show that [itex] (\mathbb{Z} _4 , \oplus ) \approx ( \mathbb{Z} ^{0}_{5} [/itex] [itex] , \odot ) [/itex]
Meaning, they are isomorphic. The 0 means the zero is deleted from the set. We are using circle plus and circle dot because we are not allowed to think of the operations as addition and multiplication yet. My trouble is with defining the operation [itex] \theta [/itex].


Homework Equations



I know [itex] \theta [/itex] must be operation preserving, that is, [itex] \theta (x \oplus y) = \theta (x) \odot \theta (y) [/itex]

The Attempt at a Solution



I tried defining [itex] \theta [/itex] as [itex] \theta ([x])= [x+1] [/itex]
so, [itex] \theta ([0]) = [1] [/itex] up to [itex] \theta ([3]) = [4] [/itex]
And now, [itex] \theta ([x] \oplus [y]) = \theta ([x \oplus y]) = [x \oplus y +1] [/itex]
Did I make the wrong steps there? I'm not sure how to arrive at [itex] \theta ([x]) \odot \theta ([y]) [/itex]
 
Last edited:

Answers and Replies

  • #2
I like Serena
Homework Helper
6,577
176
Hi Arcana!

θ needs to map each element.
So you can define it by defining θ(0), θ(1), θ(2), and θ(3).

Start with θ(0) and then make an arbitrary choice for θ(1).
Use the preservation of the operation to deduce what the other values must be.
 
  • #3
768
4
[itex] \theta ([0])=[1] [/itex]
[itex] \theta ([1])=[2] [/itex]
[itex] \theta ([2])=[3] [/itex]
[itex] \theta ([3])=[4] [/itex]

But how do I show it's operation preserving? Do I have to show it using each number, or is there some way to do it using arbitrary elements?
 
  • #4
I like Serena
Homework Helper
6,577
176
Sorry, but this won't work.
Consider θ(1+1).
 
  • #5
I like Serena
Homework Helper
6,577
176
But how do I show it's operation preserving? Do I have to show it using each number, or is there some way to do it using arbitrary elements?
Basically what you would need to do is the show that each combination of a and b has its operation preserved.
That is, set up the entire multiplication table, map it, and verify that it matches the other entire multiplication table.

With the size of your current groups this is very doable.
 
  • #6
768
4
Okay, if that's the only way I can certainly do that. thanks for the tip-off about my mapping being wrong. I think with this information I'll be okay here. :)
 
  • #7
I like Serena
Homework Helper
6,577
176
It's not the only way.
That is, you can make a couple of shortcuts.
But I guess that is better left for a later exercise.

Note that the definition of your isomorphism says: "for any elements a and b".
This means you have to show it "for any elements a and b".
 

Related Threads on Show that this is a homomorphism

  • Last Post
Replies
3
Views
1K
Replies
3
Views
1K
  • Last Post
Replies
1
Views
4K
Replies
8
Views
318
Replies
1
Views
962
Replies
4
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
553
  • Last Post
Replies
5
Views
4K
Top