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 ?
Group of order $105$
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group-theory
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2Both of which are normal? Sure; take the cyclic group of order 105. – 2010-10-11
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3And there's a non-Abelian example too :-) – 2010-10-11