Proving the Commutativity of a Group with Abstract Algebra

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Homework Help Overview

The discussion revolves around proving a property of groups in abstract algebra, specifically the relationship between the inverse of a product of elements and the commutativity of the group.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants are examining the condition under which the equation (xy)^{-1} = x^{-1}y^{-1} holds and its implications for the group being abelian. There are attempts to clarify the definitions and properties of group elements and their inverses.

Discussion Status

The discussion includes various attempts to articulate the problem and explore the implications of the given equation. Some participants express confidence in the progress made, while others seek further clarification on the attempts and reasoning involved.

Contextual Notes

There appears to be some repetition in the posts, indicating a potential lack of clarity or consensus on the initial problem statement and its requirements.

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Let G be a group. Show (xy)^{-1} = x^{-1}y^{-1} for all x, g \in G if and only if G is abelian.


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The Attempt at a Solution

 
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I think you're done :)
 
stihl29 said:
Let G be a group. Show (xy)^{-1} = x^{-1}y^{-1} for all x, g \in G if and only if G is abelian.


Homework Equations





The Attempt at a Solution


What have you tried?
 
The element (xy)^{-1} is the inverse element of xy and therefore (xy)^{-1}xy=e where e is the identity, so...
 

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