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It is clear that a finte group of order $105$ is not simple since it contains a normal Sylow $7-$subgroup or a normal Sylow $5-$subgroup. I wonder if it is possible that there exists a Sylow $7-$subgroup and a Sylow $5-$subgroup ?

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    Both of which are normal? Sure; take the cyclic group of order 105.2010-10-11
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    And there's a non-Abelian example too :-)2010-10-11

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