The restrictions imposed by anomaly cancellation on the chiral fermion
content of nonsupersymmetric gauge theories based on various groups are studied
in spacetime dimension D=6, 8, and 10. In particular, we show that the only
mathematically consistent chiral SU(5) theory in D=6 contains three
nonidentical generations.Comment: 15 pages, revtex. v2: references added to match published versio