【Basic Abstract Algebra】Exercises for Section 2.5 — Dihedral groups

Write down the dihedral group \(D_5\).
Solution: \(D_5=\lang r,s\mid s^2=r^5=1,~srs=r^{-1}\rang\), where \(r=(12345),~s=(15)(24)\), i.e. \(D_5=\{\text{id},s,r,r^2,r^3,r^4,rs,r^2s,r^3s,r^4s\}\). We have

Thus

Prove that \(D_n\) is a proper subgroup of \(S_n\) for \(n>3\).
Proof: By Thm2.5.2, we have \(D_n\subseteq S_n\) and \(| D_n |=2n,~|S_n|=n!~(n\ge 3)\). If \(D_n\) is a proper subgroup of \(S_n\), then \(n!>2n\Rightarrow (n-1)!>2\Rightarrow n>3\).

Show that \(r^ks=sr^{-k}\).
Proof: mathematical induction.