Invariance groups of transformations of basic hypergeometric series

We show that certain two-term transformation formulas between basic hypergeometric series can easily be described by means of invariance groups. For the transformations of nonterminating (3)phi(2) series, and those of terminating balanced (4)phi(3) series, these invariance groups are symmetric groups. For transformations of (2)phi(1) series the invariance group is the dihedral group of order 12. Transformations of terminating (3)phi(2) series are described by means of some subgroup of S-6, and finally the invariance group of transformations of very-well-poised nonterminating (8)phi(7) series is shown to be isomorphic to the Weyl group of a root system of type D-5. (C) 1999 American Institute of Physics. [S0022- 2488(99)00512-5].