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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. – Qiaochu Yuan Oct 11 '10 at 17:55
And there's a non-Abelian example too :-) – Robin Chapman Oct 11 '10 at 18:10

2 Answers 2

Of course there exist Sylow $7$-subgroups and Sylow $5$-subgroups: Sylow's Theorems tell you that the group has Sylow $p$-subgroups for every $p$ (with the Sylow $p$-subgroup being trivial if $p$ does not divide $105$).

If you meant to ask "Can the group have normal Sylow $5$- and Sylow $7$ subgroup?" then the answer is "yes": the (unique) abelian group of order $105$ will do (since $105=3\times 5\times 7$ is square free, the only abelian group of order $105$ is the cyclic one).

In general, you can always find a finite group of order $n$ in which all Sylow subgroups are normal: any abelian group of order $n$, and there is always at least one, will do. If any of the primes in the factorization of $n$ is raised to a degree greater than $2$, then you can also find a nonabelian group in which all the Sylow subgroups are normal, just by taking the direct product of $p$-groups of the correct size, and picking a nonabelian one when the prime is raised to a sufficiently large degree.

And in this case, you can get a nonabelian group in which the Sylow $7$- and the Sylow $5$-subgroups are normal by first constructing a nonabelian group of order $21$ (there are no nonabelian groups of orders $15$ or $35$) which has a normal subgroup of order $7$, and then taking the direct product with with $C_5$.

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For any group of order 105, there always exists a normal sylow 7-subgroup and a normal sylow 5-group.

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