The Brauer–Clifford group of G-rings

In previous work, the author introduced the Brauer–Clifford group of certain G-algebras. This group is useful because to every irreducible character of a normal subgroup of a finite group, one can associate a unique element of a specific Brauer–Clifford group, and this element controls the Clifford theory of this character in its ambient group. In the present paper, we define the Brauer–Clifford group of G-rings. This new definition only requires us to discuss tensor products over the underlying G-ring, and it is simpler than the earlier one. We prove that the new definition yields a group which is canonically isomorphic to the Brauer–Clifford group of a corresponding suitable G-algebra.