TITLE: Showing that one axiom of the concept group action implies another (similar to group homomorphisms)
QUESTION [5 upvotes]: [Beware that my question is at the bottom of all this text.]
We defined in a group theory course a group action to be a mapping $$\nu:G\times X \rightarrow X,$$ $(G,\cdot)$ group and $X$ some set, such that the axioms $$ \textrm{(GA1)} \ \ \  \nu(1,x)=x$$ and $$ \textrm{(GA2)} \ \ \ \nu(g,\nu(h,x))=\nu(g h, x)$$
hold for every $g,h\in G$ and $x\in X$. 
We defined a group homomorphism to be a map $$f:G\rightarrow H,$$ such that the axioms $$\textrm{(HOM1)} \ \ \  f(1_G)=1_H $$  and $$\textrm{(HOM2)} \ \ \ f(gh)=f(g)f(h)$$ hold for all $g,h \in G$.
Now, obviously, these axioms for the homomorphism aren't minimal, meaning (HOM2) $\Rightarrow$ (HOM1). So, naturally, I asked myself, if the same doesn't happen with the group action axioms, meaning (GA2) $\Rightarrow$ (GA1).
Now the thing is, I can show that this is the case by showing that a mapping $\nu$  satisfies (GA2) iff the mapping $$\tau: G\rightarrow X_{\leftrightarrow} ,$$ where $X_{\leftrightarrow}$ is the group  of all bijections on $X$ together with the composition, defined by $$\tau(g)(x)=\nu(g,x),$$ is a homomorphism:
$(\tau(g)\circ \tau(h))(x)=\tau(g)(\nu(h,x))=\nu(g,\nu(h,x)),$ which by (GA2) equals $\nu(gh,x)=\tau(gh)(x)$, and $\nu(g,\nu(h,x))=\tau(g)(\tau(h)(x))$ which by (HOM2) equals $\tau(gh)(x)=\nu(gh,x)$ for all $g,h\in G$ and $x\in X$.
From that I can now show (GA1) by the following implications:
$$ \textrm{(GA2)} \Leftrightarrow \textrm{(HOM2)} \Rightarrow \textrm{(HOM1)}  \Rightarrow \textrm{(GA1)}. $$
Now (finally) my question:  Can't I show that (GA1) follows from (GA2) directly ? 
I tried doing that, but couldn't succeed, because unlike showing  $ \textrm{(HOM2)} \Rightarrow \textrm{(HOM1)}  $ I don't have an thing like an "inverse to $\nu(1,x)$" that I can apply upon $\nu(1,\nu(1,x))=\nu(1,x)$ to get $\nu(,x)=1$.

REPLY [3 votes]: Let $G=\{1_G\}$ be the trivial group, and let $X=\mathbb{Z}$. Define $\nu:G\times X\to X$ by $$\nu(1_G,n)=\begin{cases}
n,&n\text{ is even}\\
n-1,&n\text{ is odd}\;.
\end{cases}$$
Clearly (GA$1$) isn’t satisfied, but for any $n\in\mathbb{Z}$ we have $$\nu(1_G,\nu(1_G,n)) = \nu(1_G,n) = \begin{cases}
n,&n\text{ is even}\\
n-1,&n\text{ is odd}\;,
\end{cases}$$ so (GA$2$) is satisfied.