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Solved: Suppose F1 and F2 are distinct reflections in a
Chapter 2, Problem 42E(choose chapter or problem)
QUESTION:
Suppose \(F_1\) and \(F_2\) are distinct reflections in a dihedral group \(D_n\) such that \(F_{1}F_{2} = F_{2}F_{1}\). Prove that \(F_{1}F{2} = R_{180}\).
Questions & Answers
QUESTION:
Suppose \(F_1\) and \(F_2\) are distinct reflections in a dihedral group \(D_n\) such that \(F_{1}F_{2} = F_{2}F_{1}\). Prove that \(F_{1}F{2} = R_{180}\).
ANSWER:Step 1 of 2
It is given that the distinct reflections \(F_{1}\) and \(F_{2}\) lie in dihedral group \(D_n\), such that:
\(F_{1}F_{2} = F_{2}F_{1}\)
It is known that in any dihedral group: \(F_{j}F_{k} = R_{i}\) and \(F_{j} = F_{j}^{-1}\)
So we can say that at some point \(i\):
\(\begin{array}{l} F_{i} F_{i}=F_{i} F_{i}^{-1} \\ F_{i} F_{i}=I \end{array}\)