TITLE: Group of order $2n$ and subgroups of order $n$
QUESTION [6 upvotes]: From a French oral examination (second year) :

Let $G$ be a finite group of order $2n$. Show that the number of subgroups of $G$ of order $n$ is never $2$. 

I have no ideas !

REPLY [4 votes]: Assume not,
Then $H,K$ be subgroup of index $2$, hence normal. Thus, $H\cap K$ is normal in $G$ and $|H\cap K|$ has index $4$ in $G$.
Thus, $G/H \cap K\cong Z_4$ or $G/H \cap K\cong Z_2 \times Z_2$, In firs case $G$ has a uniq subgroup of index $2$ including $H\cap K$, in second case $G$ has a $3$ subgroup of index $2$ including $H\cap K$. Contradiction.