Let $G$ be a domain in $\mathbb{C}^{n}$ and let $Aut(G)$ be the group of biholomorphic selfmaps of $G$. $G$ is called:
(1) homogeneous if $Aut(G)$ acts transitively on $G$, i.e. for any $z,w\in G$ there is $\varphi\in Aut(G)$, such that $\varphi(z)=w$;
(2z) symmetric at $z\in G$ if there is $\varphi\in Aut(G)$, such that $\varphi\circ\varphi =id$ and $z$ is an isolated fixed point of $\varphi$;
(2) symmetric if it is symmetric at any of its points.
It seems that (2) is equivalent to (1) and (2z) for some $z\in G$, but in many cases (1) alone implies (2).
Q1: Is there an elementary proof, that (2) implies (1)?
Additionally consider the following two properties:
(3z) for any $w\in G$ there there is $\varphi\in Aut(G)$, such that $\varphi(z)=w$ and $\varphi(w)=z$;
(3) for any $z,w\in G$ there there is $\varphi\in Aut(G)$, such that $\varphi(z)=w$ and $\varphi(w)=z$.
Clearly, (3z) for some $z\in G$ implies (1), and then using (1) in addition to (3z), one can show (3).
Also, using the classification of the bounded symmetric domains, one can show that (2) implies (3).
Q2: Does the property (3) have a name? Was it considered in the literature? Does (1) imply (3) back, or does (3z) follow from (2z) for some $z\in G$? Is there an elementary proof that (2) implies (3)?
Thank you.
