College Algebra Exam Review 244

College Algebra Exam Review 244 - G .P/ W PŁ D j X j ŒN G...

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254 5. ACTIONS OF GROUPS Corollary 5.4.10. (Second Sylow theorem). Let P and Q be two p –Sylow subgroups of a finite group G . Then P and Q are conjugate subgroups. Proof. According to the theorem, there is an a 2 G such that aQa ± 1 ± P . Since the two groups have the same size, it follows that aQa ± 1 D P n Theorem 5.4.11. (Third Sylow theorem). Let G be a finite group and let p be a prime number. Let p n be the order of a p –Sylow subgroup of G . The number of p –Sylow subgroups of G divides j G j =p n and is congruent to 1 . mod p/ . Proof. Let P be a p –Sylow subgroup. The family X of p –Sylow sub- groups is the set of conjugates of P , according to the second Sylow theo- rem. Let P act on X by conjugation. If Q is a p –Sylow subgroup distinct from P , then Q is not fixed under the action of P ; for if Q were fixed, then P ± N G .Q/ , and by Exercise 5.4.3 , P ± Q . Therefore, there is exactly one fixed point for the action of P on X , namely, P . All the nonsingleton orbits for the action of P on X have size a power of p , so j X j D mp C 1 . On the other hand, G acts transitively on X by conjugation, so j X j D ŒG W N G .P/Ł . But then ŒG W D ŒG W N G .P/ŁŒN
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Unformatted text preview: G .P/ W PŁ D j X j ŒN G .P/ W PŁ; so j X j divides j G j =p n . n We can summarize the three theorems of Sylow as follows: If p n is the largest power of a prime p dividing the order of a finite group G , then G has a subgroup of order p n . Any two such subgroups are conjugate in G and the number of such subgroups divides j G j and is conjugate to 1 . mod p/ . Example 5.4.12. Let p and q be primes with p > q . If q does not divide p ² 1 , then any group of order pq is cyclic. If q divides p ² 1 , then any group of order pq is either cyclic or a semidirect product Z p Ì ˛ Z q . In this case, up to isomorphism, there is exactly one nonabelian group of order pq . Proof. Let G be a group of order pq . Then G has a p –Sylow subgroup P of order p and a q –Sylow subgroup of order q ; P and Q are cyclic, and since the orders of P and Q are relatively prime, P \ Q D f e g . It follows...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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