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I didn't try to log into your site actually, but did look at your page. It looks like you still have the JomSocial Facebook Connect functionality still enabled. You'll need to disable this to prevent some conflicts, like you're seeing. You can do this by removing your app/secret keys from the Facebook Connect tab in the JomSocial configuration area. You'll also want to disable the JomSocial Facebook Connect plugin.

The reason I say this is because if you connect through JomSocial's FB Connect, JFBConnect won't fire like it should and do it's profile import. Additionally, since you're using 2 Facebook Connect extensions, it confuses Facebook as to which extension Facebook should be talking to. If you look at the HTML of your page, and search for 'all.js', you'll see it in the code twice. Directly above these lines is a line that specifies your appId, which is different in both cases. This will really, really confuse Facebook and will give very random results.

Hope this helps explain the situation, and possible problems you're having!

To remove JomSocial's logout, you'd have to modify the JomSocial templates to remove it. I honestly don't know all the places that you'd have to remove this from off-hand, but could help you if you can't find them. Hopefully, there's just one file that you can edit that would remove it on all pages.

The other option is to disable the "Automatically log Facebook Users In" in the JFBConnect administration area. That way, if a user logs out using the JomSocial logout button, they won't be logged out of Facebook, but they won't be automatically logged back in.. efectively logging them out of your site.

Hope this all helps! Finally, if you use JFBConnect please consider posting a rating and a review at the Joomla! Extensions Directory:
extensions.joomla.org/extensions/social-...ook-integration/7215

Thanks, and good luck!