TITLE: A group of order 105 contains a subgroup of order 35.
QUESTION [8 upvotes]: I have to prove  that a group of order 105 contains a subgroup of order 35. Could anybody tell me how to prove this?
Thanks.

REPLY [5 votes]: If $G$ is a group of order $pqr$, where $p < q < r$ are primes, then $G$ contains a normal subgroup $R$ of order $r$ (for hints on how to prove this, see this question). Since $G$ also contains a subgroup $Q$ of order $q$, the subgroup $QR$ has order $qr$. Your problem is the case $p = 3$, $q = 5$ and $r = 7$.