 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/b2/a/solution

The orbit of

s

O_{s} = {g s : g \in G}

, and its stabilizer is

G_{s} = {g \in G : g s = s}

. The identity fixes

s

; if

g, h

fix it then

(g h) s = g (h s) = s

; and

g s = s

implies

g^{- 1} s = s

. Thus $G_{s}$ is a subgroup.

A group action is transitive when it has just one orbit: for every

s, t \in S

there is

g \in G

with

g s = t

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