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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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for this question 9.

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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