0

I am studying for my final in Abstract Algebra, and I need some help. Here is the question:

Define a relation on the group $S_3$ by $\sigma \sim \tau$ iff there exists a $\rho \in S_3$ such that $\tau=\rho \sigma \rho^{-1}$

Prove that $\sim$ defines an Equivalence relation on $S_3$. I know I need to show $\sim$ is reflexive, symmetric, and transitive.

Ottavio
  • 2,287
Cameron Burgin
  • 5

1 Answers1

2

It's a matter of writing up the definitions, there's nothing deep here:

  • Reflexive: $\sigma \sim \sigma$ because $\sigma = e\sigma e^{-1}$, where $e\in S_3$ is the identity.

  • Symmetric: if $\sigma\sim \tau$, there is $\rho \in S_3$ such that $\tau = \rho \sigma \rho^{-1}$, and this implies that $\sigma = \rho^{-1}\tau \rho = (\rho^{-1})\tau (\rho^{-1})^{-1}$ --- this means that $\tau \sim \sigma$.

  • Transitive: if $\sigma \sim \tau$ and $\tau \sim \theta$, there are $\rho, \upsilon\in S_3$ such that $\tau = \rho\sigma\rho^{-1}$ and $\theta = \upsilon \tau \upsilon^{-1}$ --- this implies that $\theta = \upsilon \rho\sigma\rho^{-1}\upsilon^{-1}=(\upsilon\rho)\sigma(\upsilon\rho)^{-1}$, meaning that $\sigma\sim\theta$.

The fact that you're dealing with permutations and $S_3$ is irrelevant and this is a particular instance of a general phenomenon: if $G$ is a group acting on a set $X$ and we define $\sim$ on $X$ via $x\sim x'$ if there is $g\in G$ with $x'=g\cdot x$, then $\sim$ is an equivalence relation (and $X/_{\sim}$ is denoted by $X/G$).

Ivo Terek
  • 77,665
  • 1
    Thank you so much! This class has been quite difficult for me, and your solution makes so much sense. – Cameron Burgin Dec 09 '20 at 22:47
  • You're welcome! Try to write a proof for the fact stated in the last paragraph, and compare with the solution given to the original problem, it is instructive ($S_3$ acts on itself via conjugation). – Ivo Terek Dec 09 '20 at 22:49