Algebra Solutions: A property of multiplication in Dih(2n)

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 $rx = rsr^j = sr^{-1}r^j = sr^j r^{-1} = x r^{-1}$ .

Algebra Solutions

A property of multiplication in Dih(2n)

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