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I am having trouble with this homework problem:

Let $A$ be a set and $*$ be an associative binary operation on $A$ with the identity element $e$. Let $R$ be the relation on $A$ defined as follows: Let $a$ and $b$ exist in $A$. Then $aRb$ if there exists an invertible element $c\in A$ such that $b=c^{-1}*a*c$. Prove that $R$ is an equivalence relation on $A$.

I know that in order to prove that $R$ is an equivalence relation, I must prove that $R$ is reflexive, transitive and symmetric. But I cannot figure out how to do it. Please help!

Brian M. Scott
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user69839
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1 Answers1

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I’ll get you started. Note first that $e*e=e$, so $e$ is invertible, and $e^{-1}=e$.

  • Reflexivity: Let $a\in A$; you need to show that $aRa$. This means finding an invertible $c\in A$ such that $c^{-1}*a*c=a$. What if you take $c=e$?

  • Symmetry: Suppose that $a,b\in A$ and $aRb$; you need to show that $bRa$. You know that there is a $c\in A$ such that $c^{-1}*a*c=b$. Multiply that equation on the left by $c$ and on the right by $c^{-1}$ to get $c*c^{-1}*a*c*c^{-1}=c*b*c^{-1}$. Now simplify the lefthand side, and answer this question: what is $(c^{-1})^{-1}$?

  • Transitivity: Suppose that $a,b,c\in A$, $aRb$, and $bRc$; you need to show that $aRc$. You know that there are $g,h\in A$ such that $b=g^{-1}*a*g$ and $c=h^{-1}*b*h$; can you find a way to combine these two equations to get an element $x\in A$ such that $c=x^{-1}*a*x$? Substitution, and one more fairly basic fact about inverses should get you there.

Brian M. Scott
  • 616,228