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Past exam of the mathematics course of the University of Oxford / 2014 / Prelims / Mathematics I / 7 / a / ii

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Oxford 2014 Prelims Mathematics I 7 a
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ii
The group action law gives
ϕg​∘ϕg−1​=ϕe​=idS​,ϕg−1​∘ϕg​=idS​.
(1)
Thus ϕg​ is a bijection with inverse ϕg−1​, hence a permutation of S. Also, for every s,
ϕgh​(s)=ϕ(g,ϕ(h,s))=(ϕg​∘ϕh​)(s).
(2)
Therefore the assignment Φ(g)=ϕg​ takes values in the symmetric group Sym(S) and satisfies
Φ(gh)=Φ(g)Φ(h).​
(3)
It is a group homomorphism, with the group product on the right interpreted as composition of permutations.

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