A separable deformation of the quaternion group algebra

Abstract

Abstract. The Donald-Flanigan conjecture asserts that for any finite group G and any field k, the group algebra kG can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group Q 8 over a field k of characteristic 2 was considered as a counterexample. We present here a separable deformation of kQ 8 . In a sense, the conjecture for any finite group is open again.

Separable deformations of the generalized quaternion group algebras

“…Separability of the deformed algebra [kQ 2 n ] t is proved in the same fashion as in [1] for the case n = 3. By (2.3), (2.13) and (2.16),…”

“…z(e c1 + e c2 ) t=0 = 0 (z can be taken as t m for sufficiently large m). Plugging this choice of z in (2.14) and identifying σ and x as in (1.4) Separability of the deformed algebra [kQ 2 n ] t is proved in the same fashion as in [1] for the case n = 3. By (2.3), (2.13) and (2.16), (3.1)…”

“…In [1] a separable deformation was constructed for the quaternion group G = Q 8 , turning the generalized quaternion group Q 16 to a current minimal unsolved case. In this note, we extend the strategy of [1] in order to deal, as promised therein, with the family of generalized quaternion groups Q 2 n . As a by-product we establish separable deformations for the family of dihedral 2-groups D 2 n (K. Erdmann and M. Schaps have already found separable deformations for this family in [4]).…”

“…It was verified in [3,4,5,6,7,8,11,12,13] for certain families of finite groups. In [1] a separable deformation was constructed for the quaternion group G = Q 8 , turning the generalized quaternion group Q 16 to a current minimal unsolved case. In this note, we extend the strategy of [1] in order to deal, as promised therein, with the family of generalized quaternion groups Q 2 n .…”