First Isomorphism Theorem Question

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  • #1
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I am having a hard time using or applying the theorem .

Anyways

Prove that there is no homomorphism from Z[itex]_{8}[/itex][itex]\oplus[/itex]Z[itex]_{2}[/itex] onto Z[itex]_{4}[/itex][itex]\oplus[/itex]Z[itex]_{4}[/itex]

Im guessing its the First Isomorphsim Theorem because its in the chapter. But Im not sure how to use it.
 

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  • #2
Dick
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Show the kernel of the homomorphism must be nontrivial. How might you do that?
 
  • #3
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just go through and try all the elements until you exhaust them all. Do I want to find the kernel of the first thing or the second thing?
 
  • #4
Dick
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just go through and try all the elements until you exhaust them all.
How can you do that?! You aren't given the homomorphism. Look at the orders of elements in the two groups.
 
  • #5
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Cant I just make one up?
And how do you work in the factor group?
 
  • #6
Dick
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Cant I just make one up?
And how do you work in the factor group?
You can make one up if you want to. But showing that isn't onto doesn't show another homomorphism might not be onto. If the kernel is nontrivial then it can't be onto. What's the reason for that?
 
  • #7
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So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?
 
  • #8
Office_Shredder
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So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?
I don't see any particular reason for why that would be the kernel.

Suppose that we call the kernel of the map K. Then (Z8+Z2)/K = Z4+Z4 is an isomorphism of groups by the first isomorphism theorem. What is the cardinality of Z8+Z2/K in terms of |K|? What is the cardinality of Z4+Z4?
 
  • #9
Dick
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So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?
That's pretty poorly worded. (8,0)=(0,0) isn't a true statement. Call the homomorphism h. Then why is h((8,0))=(0,0)?
 
  • #10
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(Z8+Z2)/Z4+Z4 = K

so |K| = 16/16=1 so wouldnt the kernel be the identity?
 
  • #11
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That's pretty poorly worded. (8,0)=(0,0) isn't a true statement. Call the homomorphism h. Then why is h((8,0))=(0,0)?
because 8 mod 4 is zero and 0 mod 4 is zero
 
  • #12
Dick
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because 8 mod 4 is zero and 0 mod 4 is zero
There's a kernel of truth in there. Now if you would just state it in terms that involve group theory that might be helpful. Why are you talking about mod 4?
 
  • #13
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we are mapping over to Z4
 
  • #14
Dick
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we are mapping over to Z4
No you aren't! You are mapping to Z4+Z4.
 
  • #15
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yes but component wise we are mapping each thing to Z4, is that the wrong way to look at it ?
 
Last edited:
  • #16
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Hint: One of the groups has an element of order ______ but the other one ______.
 
  • #17
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aren't both groups order 16 because 8*2 =16 and 4*4=16?
 
  • #18
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Wait Z8 has an element of order 8 and Z4 has an element of at most 4. Is that right.... if it is, i'm not sure if that is helpful
 
  • #19
Dick
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yes but component wise we are mapping each thing to Z4, is that the wrong way to look at it ?
It's the wrong way to look at it. You are throwing out numbers like you know what the homomorphism is and you are just cranking out arithmetic. You don't know what the homomorphism is. You can't do that. This is a course in group theory, not arithmetic. Group theory is an abstraction of arithmetic. Use group theory terminology to express yourself.
 
  • #20
Dick
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Wait Z8 has an element of order 8 and Z4 has an element of at most 4. Is that right.... if it is, i'm not sure if that is helpful
That's very helpful. Now use it. It's your job to figure out how to use it.
 

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