When you say "even the wifi is on" do you mean that the wifi network is on (i.e. the router) or do you mean even when you have wifi turned on in your phone?
'To see available connections tern on WiFi' (sic) is not referring to your network router being available to devices. It means turn on wifi in your phone. I apologize if this is what you meant.
by the way, you might have received more replies if you had posted this in the general S3 forum and not the All Things Root subforum.