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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.
- Fall '08