This is my third week of abstract algebra.
Let $H_1,H_2$ be two subgroups of $G$. Show that $$ G=H_1 \cup H_2 \implies G=H_1 \lor G=H_2 $$
Here is what I thought:
If we consider subgroups of $\Bbb Z/4\Bbb Z=\{0,1,2,3 \}$, then this rule would be true:
\begin{equation} \forall H_1,H_2\subseteq G :H_1\subseteq H_2 \lor H_2\subseteq H_1 \tag{1} \end{equation}
Which would prove the statement. However, if I look at the vectorspace $\Bbb R^3$, then $(1)$ wouldn't be true. For that example, one of those subgroups has to be three dimensional, does there exist something as dimensions in group theory ?
I think I'm looking in the wrong direction, can somebody enlighten me a little bit here ? A subtle hint would be appreciated.