Is Group G Abelian if f(a)=a^-1 is a Homomorphism?

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Homework Help Overview

The discussion revolves around the properties of group theory, specifically whether a group G is abelian if the function f(a) = a^-1 is a homomorphism. The original poster seeks to understand the relationship between the definitions of abelian groups and homomorphisms in this context.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the definitions of abelian groups and homomorphisms, with some expressing confusion about the terms "commutative" and "abelian" being synonymous. Others suggest starting with definitions and properties to clarify the proof process.

Discussion Status

The discussion includes attempts to clarify definitions and the relationship between the properties of the group and the function f. Some participants have provided guidance on how to approach the problem by focusing on definitions and the implications of the homomorphism condition.

Contextual Notes

There is a noted concern regarding the original poster's understanding of the terms used, particularly the equivalence of "commutative" and "abelian." The original poster expresses uncertainty about how to begin the proof.

ldelong
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Homework Statement


Prove that Group G is abelian iff the function f:G to G given by f(a)=a^-1 is a homomorphism

Homework Equations





The Attempt at a Solution



Group G must be communitative for it to be abelian I have no idea where to start to begin to prove this I know that homomorphic means that complete the operation first on a in the group gets mapped to an element that the operation is completed 2nd. Help... I really need a jumping point.
 
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You can start by writing out some relevant definitions and basic facts along with the facts you're given. Often this is also where you finish. In this case, remember that the condition for f to be a homomorphism is f(ab)=f(a)f(b). Now use the definition of f.
 
Last edited:
ldelong said:
Group G must be communitative for it to be abelian


When I see things like that I worry. Commutative and abelian are synonyms.
 
Actually there's not much to prove. It is, as StatusX already said, a matter of writing out the definition of homeomorphism and plugging in commutativity of the group and the definition of f at the right points.

So just start writing something down, then tell us where you get stuck.
 
I got it thanks
 

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