Question: Is f: Z2 -> Z2 with f(x) = -x a homomorphism?

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Homework Help Overview

The discussion revolves around the properties of a function defined as f(x) = -x, specifically examining whether it acts as a homomorphism from Z2 to Z2. Participants are exploring the implications of this function within the context of group theory.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants attempt to establish whether the function is a homomorphism by comparing it to a similar function defined on Z. Others express uncertainty about the validity of the function when applied to Z2, questioning its properties and the implications of the group structure.

Discussion Status

The conversation is ongoing, with participants clarifying their points and attempting to address misunderstandings. Some guidance has been offered regarding the nature of homomorphisms, but no consensus has been reached on the specific case of f: Z2 -> Z2.

Contextual Notes

There is some confusion regarding the notation and definitions used, particularly with the variable 's' and its relation to the groups being discussed. Participants are also reflecting on their previous statements and attempting to clarify their reasoning.

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(A) f: Z -> Z with a function defined as f(x)=-x

1.) f(0)=0
2.) f(a)-f(b)=-a+b=-a+b=f(a-b)
3.) -f(a)=a=a=f(-a)

Conclusion: a homomorphism

(B) f: Z2 -> Z2 with a function defined as f(x)=-x
This one I'm not sure, I doubt it is a homomorphism though.

f: g -> g
 
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Z_s? s equal what? s=2 certainly is a homomorphism. s>2 may cause problems. But not with f(0).
 


Sorry, I didn't even realize all the things i left off. But anyway, are my three proofs correct?
 


What three proofs? I thought there were maximum two problems here. f(x)=-x for i) Z->Z and ii) Z_2->Z_s. Whatever s is. Homomorphism, yes or no. You are being super unclear.
 


Wow, sorry. I just transposed everything from my notes quickly. The S was suppose to be a 2, which I fixed. I also only did put up two proofs just to see if Z -> Z and Z2 -> Z2 were homomorphisms. I arrived at the same conclusion for both. I actually didn't type out the third one, I'm not sure why.
 

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