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Unformatted text preview: 5.4. GROUP ACTIONS AND GROUP STRUCTURE 255 from the third Sylow theorem that P is normal in G , since 1 is the only natural number that divides pq and is congruent to 1 . mod p/ . Therefore, PQ D QP is a subgroup of G of order pq , so PQ D G . According to Corollary 3.2.5 , there is a homomorphism W Z q ! Aut . Z p / such that G Z p Z q . Since Z q is simple, is either trivial or injective; in the latter case, . Z q / is a cyclic subgroup of Aut . Z p / of order q . But, by Corollary 5.3.4 , Aut . Z p / Z p 1 . Therefore, if q does not divide p 1 , then must be trivial, so G Z p Z q Z pq . On the other hand, if q divides p 1 , then Aut . Z p / Z p 1 has a unique subgroup of order q , and there exists an injective homomorphism of Z q into Aut . Z p / . Thus there exists a nonabelian semidirect product Z p Z q . It remains to show that if and are non-trivial homomorphisms of Z q into Aut . Z p / , then Z p Z q Z p Z q . Since...
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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