Group of order $2n$ and subgroups of order $n$

HINT: Suppose that $H$ and $K$ are distinct subgroups of $G$ of order $n$, and let $L=H\cap K$. Let $D=G\setminus(H\cup K)$.

• Show that if $x\in H\setminus L$, then $x(K\setminus L)\subseteq D$.
• Conclude that $|H\setminus L|\le|L|$ and hence that $|L|=\frac{|H|}2=\frac{n}2$.
• Show that $L\cup D$ is a third subgroup of $G$ of order $n$.

You don't need this, but in fact $L$, like $H$ and $K$, is normal in $G$, and $G/L$ is (isomorphic to) the Klein $4$-group, which has three subgroups of order $2$; one is $H/L$, one is $K/L$, and one is $(L\cup D)/L$.

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.