Unformatted text preview: unique Sylow 3subgroup which must be P , and we have P ± PH . Since groups of prime and prime squared order are abelian and P , H are normal subgroups of PH with coprime order, we can now deduce that PH is abelian. Similarly QH is abelian. Let h ∈ H . Then PH , QH ≤ C G ( h ) (because PH , QH are abelian), hence 3 2 * 17 and 5 2 * 17 divide  C G ( h )  . Therefore 3825 ± ±  C G ( h ) and we deduce that C G ( h ) = G . We conclude that h ∈ Z ( G ) for all h ∈ H and it follows that H ≤ Z ( G ) ....
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This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.
 Fall '07
 PALinnell
 Math, Algebra

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