Unformatted text preview: of all x 2 X . For example, the kernel of the action of G on itself by conjugation is the center of G . Application: Counting Formulas. It is possible to obtain a number of well–known counting formulas by means of the proposition and its corollary. Example 5.1.19. The number of kelement subsets of a set with n elements is ± n k ² D nŠ kŠ.n ² k/Š : Proof. Let X be the family of k element subsets of f 1;2;:::;n g . S n acts transitively on X by ± f a 1 ;a 2 ;:::;a k g D f ±.a 1 /;±.a 2 /;:::;±.a k / g . (Verify!) The stabilizer of x D f 1;2;:::;k g is S k ± S n ± k , the group of permutations that leaves invariant the sets f 1;2;:::;k g and f k C 1;:::;n g . Therefore, the number of kelement subsets is the size of the orbit of x , namely, j S n j j S k ± S n ± k j D nŠ kŠ.n ² k/Š :...
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 Fall '08
 EVERAGE
 Algebra, Group Theory, Sets, Normal subgroup, Z4, Z4 Z4 Z4

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