TITLE: May a group have two disjoint subgroups?
QUESTION [10 upvotes]: Is there any group $(G,+)$, that has $2$ or more subgroups $(H,+),\; (I,+)$, where 
$$H \cap I = \emptyset?$$
I believe not, because both $H$ and $I$ must contain neutral element.
Is my assumption true?

REPLY [5 votes]: Let's try to prove this out of the axioms of a group. We know that $G,H,I$ are all groups under $+$ and because $H,I$ are subgroups we know that $H,I\subseteq G$.
Let $h\in H$ and $i\in I$, then $h,i\in G$. By defintion there exist an element $e\in G$, such that
\begin{align}
&e+h=h \tag 1\\
&e+i=i \tag 2
\end{align}
But as $H,I$ are also groups, there exist also an element $e_h\in H,e_i\in I$ such that
\begin{align}
&e_h+h=h \tag 3\\
&e_i+i=i \tag 4
\end{align}
By definition of a group, $h$ and $i$ both have a left-inverse, therefore we obtain:
\begin{align}
&e+h\overset{1}{=}h\overset{3}{=}e_h+h\implies e=e_h \\\\
&e+i\overset{2}{=}i\overset{4}{=}e_i+i\implies e=e_i 
\end{align}
This implies your result:
$$\emptyset\not=\{e\}\subseteq H\cap I$$