Algebra Solutions: Exhibit an alternate presentation for Dih(4)

Exhibit an alternate presentation for Dih(4)

Show that $D_4 = \langle x, y \ |\ x^2 = y^2 = (xy)^2 = 1 \rangle$ using the substitutions $x = r$ and $y = s$ .

It suffices to show that $(xy)^2 = 1$ implies $rs = sr^{-1}$ and vice versa. For the forward direction, we have $(rs)^2 = rsrs = 1$ so that $rs = s^{-1} r^{-1} = sr^{-1}$ . For the backward direction, we have $rs = sr^{-1}$ so that $rsrs^{-1} = (rs)^2 = (ab)^2 = 1$ .

Algebra Solutions

Exhibit an alternate presentation for Dih(4)

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