A property of multiplication in Dih(2n)


Recall that D_{2n} = \langle r,s \ |\ r^n = s^2 = 1, rs = sr^{-1} \rangle.
Use the presentation of D_{2n} to show that if x \in D_{2n} is not a power of r then rx = xr^{-1}.

Every element x \in D_{2n} is of the form x = s^i r^j where i = 0,1 and 0 \leq j < n. If i = 0 we have that x is a power of r; thus x = sr^j for some 0 \leq j < n. Hence rx = rsr^j = sr^{-1}r^j = sr^j r^{-1} = x r^{-1}.





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