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The permalink for the registration page was set by you to "register-2"

13008

I have now changed that back to "register" and everything works fine. I also checked if the email comes.

Please...

I think i found allready the reason (without the need of re-installing)

You had set in "Front page displays" => "Posts page:" to Categories instead of Blog

I have changed that back and now you...

I'll need admin access to check your install.

Re-set your permalinks. Try default or postname.