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Let G be a nite gmup with identity element e. Let 15' act en a nite set X. This means that there is...

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9. Let G be a finite gmup with identity element e. Let 15‘ act en a finite set X. This means that there is
amapGKX—+X,theaetian._suehthat[9h]:=g{hz}andem=zwheneg1hEGandeXThe
orbit efp is the subset an given by Gp= {g'p I gE G}. For each 131-bit1 let 3:11p} he the stabilizer nip
definedhy3t{p}={gEGIgp=g}. Let Xthetheeetaferbite. Let D. be the Elihedra] group with eight elements generated by G —l l [I
1 fl ’ fl —1
Let X = {(:l:l,l]}1 {IL :tl'jl}, the vertiees nfa equate. Enmpute G[l.,fl-}. Compute Stfili fl}. Prove the tulle-wing general facts:
Prove that Ship} is a subgroup of G. Show that |Gp| = leSttpn. Showthat
|X|= E IGESEIPH apex-m

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