Is G an Abelian Group Given Specific Conditions?

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Discussion Overview

The discussion revolves around whether a group \( G \) can be shown to be an Abelian group under specific conditions involving the equation \( ayb = cyd \) for elements \( a, b, c, d, y \in G \). The context is primarily theoretical and homework-related, focusing on group properties and implications.

Discussion Character

  • Homework-related, Mathematical reasoning

Main Points Raised

  • Post 1 introduces the problem of proving that \( G \) is an Abelian group given the condition \( ayb = cyd \) implies \( ab = cd \).
  • Post 2 suggests considering the case where \( y \) is the inverse of an element.
  • Post 3 questions how to ensure \( ayb = cyd \) holds true when assuming \( y \) is an inverse.
  • Post 4 reiterates the uncertainty about ensuring \( ayb = cyd \) is valid and emphasizes the need to choose \( y, c, \) and \( d \) appropriately to satisfy the equation.
  • Post 5 presents a specific case where \( c = b \), \( d = a \), and \( y = a^{-1} \), leading to the conclusion that \( ab = ba \), thus supporting the claim that \( G \) is an Abelian group.
  • Post 6 confirms the correctness of Post 5's reasoning.

Areas of Agreement / Disagreement

While there is a progression towards a solution, the discussion includes uncertainty and questions about the validity of certain assumptions. Post 5's conclusion is affirmed by Post 6, indicating some level of agreement, but earlier posts express doubts about the implications of the conditions.

Contextual Notes

The discussion includes assumptions about the choice of \( y, c, \) and \( d \) that are not fully resolved, as well as the dependence on the specific conditions provided in the problem statement.

alexmahone
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Let $G$ be a group such that for all $a$, $b$, $c$, $d$, and $y\in G$ if $ayb=cyd$ then $ab=cd$. Show that $G$ is an Abelian group.

HINTS ONLY as this is an assignment problem.
 
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Consider $ayb=cyd$ when $y$ is the inverse of something.
 
Evgeny.Makarov said:
Consider $ayb=cyd$ when $y$ is the inverse of something.

Even if I suppose $y=a^{-1}$ or $b^{-1}$ or $a^{-1}b^{-1}$, how do I know that $ayb=cyd$ has to be true?
 
Alexmahone said:
Even if I suppose $y=a^{-1}$ or $b^{-1}$ or $a^{-1}b^{-1}$, how do I know that $ayb=cyd$ has to be true?
Given $a$ and $b$, you can make $ayb=cyd$ true by choosing $y$, $c$ and $d$ appropriately.

Another way to look at this is the following. You need to prove $ab=ba$ for all $a$ and $b$. Try to apply the implication that is given to you in post #1. For this you have to guess $y$ because it occurs only in the assumption and not the conclusion.
 
I think I got it!

Take $c=b$ and $d=a$
Take $y=a^{-1}$

$ayb=aa^{-1}b=b$
$cyd=ba^{-1}a=b$
So, $ayb=cyd$
$\implies ab=cd$
i.e. $ab=ba$
So, $G$ is an Abelian group.
 
Yes, that's correct.
 

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