# Composition of elements in dihedral group

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?

• That's not even true in the example you've given! – Lord Shark the Unknown May 16 at 16:17
• Ahhh sorry just fixed it! @LordSharktheUnknown – Mathsical Studies May 16 at 16:18
• 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 $n$... – dan_fulea May 16 at 16:20
• Sorry just fixed! @dan_fulea – Mathsical Studies May 16 at 16:21
• 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 $$r^i s r^j =sr^{-i}\ r^j\ .$$ – dan_fulea May 16 at 16:28

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 $$D_6$$) $$\rho^3=\rho^{-3}$$.