Is the axiom $g1 = g$ essential for a group action A group $G$ acts on a set $\Omega$ if 
(1) $\omega\cdot 1_G = \omega$
(2) $(\omega \cdot g)\cdot h = \omega \cdot (gh)$
for all $\omega \in \Omega$ and $g,h \in G$. But is (1) really essential, what if we drop (1) and just require (2), then we must have
$$
 \omega\cdot 1_G = \omega \cdot (gg^{-1}) = (\omega\cdot g)\cdot g^{-1}
$$
for all $g \in G$.

So do you know any example where (2) holds, but (1) fails?


Let $\Omega = \{1,2\}$, take $G$ to be any group and set $x \cdot g = 1$ for all $x \in \Omega$ and $g \in G$. Then of course $(x \cdot g) \cdot h = 1 \cdot h = 1 = x \cdot (gh)$, but $2 \cdot 1_G = 1 \neq 2$ .
