Concerning the situation in characteristic $p$: when $p$ divides the order of $G$, the case not covered by Maschke's theorem, the group algebra $KG$ is no longer semisimple. There is, whoeverhowever, a whole array of papers, started from J.D. Donald and F.J. Flanigan, A deformation theoretic version of Maschke's theorem for modular group algebras: the commutative case, J. Algebra 29 (1974), 98-102, DOI:10.1016/0021-8693(74)90114-8, aiming to prove a conjecture which can be considered as a modular analog of Maschke's theorem: the group algebra KG is deformed to a semisimple algebra. Most of these papers have a group-theoretic flavor, arguing in terms of blocks and other group-representation-theoretic data. Murray Gerstenhaber and Anthony Giaquinto have claimed in: Compatible deformations, Trends in the Representation Theory of Finite Dimensional Algebras (ed. E.L. Green and B. Huisgen-Zimmerman), Contemp. Math. 229 (1998), 159-168, that there is a counterexample to this conjecture: a 8-element quaternion group over a field of characteristic 2. This was believed to be true for a decade or so, after it has been proved wrong (N. Barnea and Y. Ginosar, A separable deformation of the quaternion group algebra, Proc. Amer. Math. Soc. 136 (2008), 2675-2681, DOI: 10.1090/S0002-9939-08-09480-X, arXiv:0704.1556). As far as I know, the Donald-Flanigan conjecture is still open.
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Concerning the situation in characteristic $p$: when $p$ divides the order of $G$, the case not covered by Maschke's theorem, the group algebra $KG$ is no longer semisimple. There is, whoever, a whole array of papers, started from J.D. Donald and F.J. Flanigan, A deformation theoretic version of Maschke's theorem for modular group algebras: the commutative case, J. Algebra 29 (1974), 98-102, DOI:10.1016/0021-8693(74)90114-8, aiming to prove a conjecture which can be considered a modular analog of Maschke's theorem: the group algebra KG is deformed to a semisimple algebra. Most of these papers have a group-theoretic flavor, arguing in terms of blocks and other group-representation-theoretic data. Murray Gerstenhaber and Anthony Giaquinto have claimed in: Compatible deformations, Trends in the Representation Theory of Finite Dimensional Algebras (ed. E.L. Green and B. Huisgen-Zimmerman), Contemp. Math. 229 (1998), 159-168, that there is a counterexample to this conjecture: a 8-element quaternion group over a field of characteristic 2. This was believed to be true for a decade or so, after it has been proved wrong (N. Barnea and Y. Ginosar, A separable deformation of the quaternion group algebra, Proc. Amer. Math. Soc. 136 (2008), 2675-2681, DOI: 10.1090/S0002-9939-08-09480-X, arXiv:0704.1556). As far as I know, the Donald-Flanigan conjecture is still open. |
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