THE KERNEL OF A BLOCK OF A GROUP ALGEBRA

Avoiding the theory of characters of finite groups and group algebras over fields of characteristic zero a ring theoretical proof is given for R. Brauer's theorem which asserts that the (modular) kernel of a block of a group algebra FG of a finite group over a field F of characteristic p>O is a p-nilpotent normal subgroup of G. If G is a finite group and if K is a splitting field for G which has characteristicp>O, then by R. Brauer [2] the intersection N of the kernels of the irreducible representations of the group G belonging to a block B of the group algebra KG is a p-nilpotent normal subgroup of G. Furthermore, if B is the principal block of KG, then N is the maximal p-nilpotent normal subgroup of G by Theorem 2 of R. Brauer [1]. Brauer's proofs use the theory of blocks of characters of G. In this note both results are proved for group algebras over arbitrary fields using only elementary ring theoretical methods. Throughout this note FG denotes the group algebra of the finite group G over the field F with characteristic p>O, and J(FG) is the Jacobson radical of FG. A block of a group algebra FG (in the sense of A. Rosenberg [6]) is a triple B-e-A consisting of a minimal direct (two-sided) summand B of FG, its identity element e, and a linear character A of the center ZFG belonging to e. B

Keywords:

• Finite group
• Center (group theory)
• Ring (mathematics)
• Kernel (algebra)
• block
• Group algebra
• Normal subgroup
• Mathematics
• Group (mathematics)
• Combinatorics

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