Nitpicking: T-Mobile is the international mobile branch of Deutsche Telekom, not just in the US (though there is a US subsidiary).

Other branches include T-Online (private ISP, actually a former subsidiary), T-Systems (subsidiary proving services to the public sector and larger corporations) and T-Home (which I have trouble telling apart from T-Online). There may be other branches too, but in practice most people in Germany just lump them all together anyway.