Group

$G$ is a group if:

1. $\exists e\in G:\forall g\in G,eg=ge=g$
2. $\forall g\in G\exists g^{-1}:g{g}^{-1}=g^{-1}g=e$
3. $g_1{g}_2\in G\forall g_1,g_2\in G$
4. $(g_1{g}_2)g_3=g_1(g_2{g}_3)\forall g_1,g_2,g_3\in G$