Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 1 / 1B / a

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- Solution a

Solution

For a finite group

G

acting on a set

X

, write

G_{x} = {g \in G : gx = x}

for the stabilizer subgroup and

G x = {gx : g \in G}

for the orbit of

x

. The orbit-stabilizer theorem gives

∣ G ∣ = ∣ G_{x} ∣ ∣ G x ∣.

Indeed,

g G_{x} \mapsto gx

is a bijection from the left cosets of

G_{x}

onto

G x

: two images agree precisely when

h^{- 1} g \in G_{x}

. Each coset has

∣ G_{x} ∣

elements, proving the formula.

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