proof of theorem SG1

Theorem SG1
If G is a group of transformations of a set S, and F is a subset of S, then
S(F,G) is a subgroup of G.

(1) Since e, the identity of G, has e(F) = F, eεS(F,G).
(2) For gεS(F,G), g(F) = F, so g-1(F) = F and hence g-1εS(F,G).
(3) For g and hεS(F,G), g(F) = F and h(F) = F, so that we have
goh(F) = g(h(F)) = g(F) = F. Then gohεS(F,G).

Together, these show that we have a subgroup of G.

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