Unformatted text preview: 1 through a p ± 1 can be chosen arbitrarily, and a p D .a 1 a 2 :::a p ± 1 / ± 1 . Thus the cardinality of X is j G j p ± 1 . Recall that if a;b 2 G and ab D e , then also ba D e . Hence if .a 1 ;a 2 ;:::;a p / 2 X , then .a p ;a 1 ;a 2 ;:::;a p ± 1 / 2 X as well. Hence, the cyclic group of order p acts on X by cyclic permutations of the sequences. Each element of X is either ﬁxed under the action of Z p , or it belongs to an orbit of size p . Thus j X j D n C kp , where n is the number of ﬁxed points and k is the number of orbits of size p . Note that n µ 1 , since .e;e;:::;e/ is a ﬁxed point of X . But p divides j X j ´ kp D n , so X has a ﬁxed point .a;a;:::;a/ with a ¤ e . But then a has order p . n Theorem 5.4.7. (First Sylow theorem). Suppose p is a prime, and p n divides the order of a group G . Then G has a subgroup of order p n . 1 J. H. McKay, Amer. Math. Monthly 66 (1959), p. 119....
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 Fall '08
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 Algebra, Group Theory, Natural number, Abelian group, Subgroup, Cyclic group

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