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I've come across the following example: $$ρ^3·σρ^2 = ρ^2σρ^{−1}ρ^2 = ρ^2σρ = ρσρ^{-1}ρ = ρσ = σρ^{−1} = σρ^5$$

And was wondering if it is true in general that $ρ^i·σρ^j = σρ^{i+j}$?

I know that $ρσ = σρ^{n-1}$ for the dihedral group $D_n$ so maybe that would help some how?

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  • $\begingroup$ That's not even true in the example you've given! $\endgroup$ – Lord Shark the Unknown May 16 at 16:17
  • $\begingroup$ Ahhh sorry just fixed it! @LordSharktheUnknown $\endgroup$ – Mathsical Studies May 16 at 16:18
  • $\begingroup$ Please fix some $n$, some dihedral group $D_n$, best give its order to eliminate conventions, and state an equality with some other letters, avoid $n$... $\endgroup$ – dan_fulea May 16 at 16:20
  • $\begingroup$ Sorry just fixed! @dan_fulea $\endgroup$ – Mathsical Studies May 16 at 16:21
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    $\begingroup$ From $rs = sr^{-1}$ in some group we have immediately $r^{-1}s = sr$ and thus $$r^i s r^j = r^i\ r^{-1}sr^{j-1}=\dots=r^i\ r^{-j}s$$ and same $$r^i s r^j =sr^{-i}\ r^j\ .$$ $\endgroup$ – dan_fulea May 16 at 16:28
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No, but what is true in general is $\rho^m \sigma \rho^k = \sigma \rho^{k-m}$.

This happens to correspond to $\rho^{k+m}$ in your specific example because (it appears you are working in $D_6$) $\rho^3=\rho^{-3}$.

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  • $\begingroup$ This is perfect, thank you! $\endgroup$ – Mathsical Studies May 16 at 16:28

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