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College Algebra Exam Review 245

College Algebra Exam Review 245 - 255 5.4 GROUP ACTIONS AND...

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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 Š Z p Ì ˛ 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 / Š Z p 1 . Therefore, if q does not divide p 1 , then ˛ must be trivial, so G Š Z p Z q Š Z pq . On the other hand, if q divides p 1 , then Aut . Z p / Š 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 Ì ˛ Z q . It remains to show that if ˛ and ˇ are non-trivial homomorphisms of Z q into Aut . Z p / , then Z p Ì ˛ Z q Š Z p Ì ˇ Z q . Since ˛ and ˇ are injective, ˛. Z q / and ˇ. Z q /
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