I play the tenor sax in honors band and in one of our pieces, call tatarian dances, in the third movement it has a G below middle C. I didn`t know this note was possible on a tenor sax. Is it a music flaw? or is it possible to extend my register?

I don`t think that it is possible because there is only enough keys to go to the low Bb. But I may be wrong, after all, I`ve been playing for a year.

That note is never in existance with the tenor sax

Just play that note an octave higher.

make sure you`re reading it right, that happens to me all the time, but a G is hard to mistake.

You might mistake that note as a G if it`s perhaps a low A. Most Baris today come equiped with a low A key, and I know there are some Altos and Tenors that have it as well, (these are rare though). If it is in fact a low A, you can just finger low Bb and stick your foot inside the bell...
Either way i`m sure it`s a transcription error. The person who arranged that piece for winds didn`t know what he or she was doing... Ya.. just play it an octave up.