Degug mode and normal mode, it is impossible to save the changes but there is no error message.
nothing has changed

I've no idea then, it works fine here. Please PM backend super administrator login credentials and will take a look.

By chance, there is no parameter in GJ something to prevent the publishing groups by the owner and I have set wrong?

Don't understand the question. Publish state is for administrative purposes only. A user can't publish/unpublish their own group. The exception to this is category owners can publish/unpublish groups in their category freely.

Something is rewriting the POST URLs. For example a post on the wall sends me to the following, which isn't valid.

catgories/pluginclass?routelat=&routelng=&wall_post=&wall_reply=0&cbsecuritym3=cbm_1c75893e_701144ed_f6f91b4423246ff84785ee20603087e4

After the POST you should be sent back to the group, but something is intercepting it and stopping it before it even successfully posts to GJ. I suspect whatever is appending "routelat=&routelng=" is what is breaking the URLs.

The only things that can do this is Joomla System plugins or a CB plugin acting on GJs trigger. I recommend disabling non-core Joomla system plugins 1 by 1 until resolved.

I can't debug any further because your website is actually crashing my entire browser.