A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

Integral and p-modular semisimple deformations for p-solvable groups of finite representation type

“…Then H is a semidirect product of (a m ) by (b 1 ). The center is trivial because there is an automorphism of G carrying b' into a power of b, and every element of (b) induces a nontrivial automorphism of (a), as in the proof of (2). Since we showed in the proof of (1) that r -1 is not divisible by p, we conclude [5] In [10], we proved that any group with cyclic p-Sylow group has a pmodular semisimple deformation for any ^-sufficiently large field.…”

“…In the nonmodular case, Maschke's theorem tells us that kG is already semisimple and no deformation is needed. Donald and Flanigan [2] proved the original conjecture for commutative groups twenty years ago. The author [10] just recently proved the conjecture for groups of finite representation type, that is, groups with cyclic p-Sylow subgroup.…”

“…In [2] Donald and Flanigan conjectured a modular version of Maschke's theorem, namely that for every finite group and every sufficiently large field k, the group algebra k G has a deformation to a semisimple algebra with the same Wedderburn components as the group algebra over a field of characteristic 0. We will call such a deformation a p-modular semisimple deformation.…”

A separable deformation of the quaternion group algebra

“…First, the leading term y 2 is central since the automorphism η t is of order 2. Next, by (4.8), the free term be 1 …”

“…In their paper [1], J. D. Donald and F. J. Flanigan conjectured that any group algebra kG of a finite group G over a field k can be deformed to a semisimple algebra even in the modular case, namely where the order of G is not invertible in k. A more customary formulation of the Donald-Flanigan (DF) conjecture is by demanding that the deformed algebra [kG] t should be separable; i.e., it remains semisimple when tensored with the algebraic closure of its base field. If, additionally, the dimensions of the simple components of [kG] t are in one-to-one correspondence with those of the complex group algebra CG, then [kG] t is called a strong solution to the problem.…”