Abstract Algebra I · Group actions, orbits, and stabilizers
Symmetries acting on a square
Problem
Let $D_4=\langle r,s\mid r^4=s^2=e,\ srs=r^{-1}\rangle$ act on the four vertices $\{0,1,2,3\}$ of a square by $r\cdot i=i+1\pmod 4$ and $s\cdot i=-i\pmod 4$. Verify the action relations, find the orbit and stabilizer of $0$, and check orbit-stabilizer numerically.
Hint
Compute $r^4(i)$, $s^2(i)$, and $srs(i)$ modulo $4$.
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