Unformatted text preview: 260 5. ACTIONS OF GROUPS homomorphism is the intersections of the stabilizers of the points of X , and, in particular, is contained in G , which is the stabilizer of G D eG 2 X . On the other hand, the kernel is a normal subgroup of S 5 . Since G does not contain S 5 or A 5 , it follows from the fact just mentioned that the kernel is f e g . But this means that S 5 is isomorphic to a subgroup of Sym .X/ Š S d , and, consequently, 5Š dŠ , or 5 d . n Remark 5.5.2. More, generally, A n is the only nontrivial normal subgroup of S n for n 5 . So the same argument shows that a subgroup of S n that is not equal to S n or A n must have index at least n in S n . Now, suppose that G is a transitive subgroup of S 5 , not equal to S 5 or A 5 . We know that for any finite group acting on any set, the size of any orbit divides the order of the group. By hypothesis, G , acting on f 1;:::;5 g , has one orbit of size 5, so 5 divides j G j . But, by the lemma, j G j 24 . Thus, j G j D 5k; where 1 k 4:...
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 Fall '08
 EVERAGE
 Algebra, Group Theory, Normal subgroup, Equivalence relation, Cyclic group, Š Sd

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