Structure of blocks with normal defect and abelian p′ inertial quotient

David Benson, Radha Kessar, Markus Linckelmann

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Abstract

Let k be an algebraically closed field of prime characteristic p. Let kGe be a block of a group algebra of a finite group G, with normal defect group P and abelian p' inertial quotient L. Then we show that kGe is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order p3 with a quaternion group of order eight with the centre acting trivially. In the case of p = 3, we give explicit generators and relations for the basic algebra as a quantised version of kP. As a second example, we give explicit generators and relations in the case of a group of shape 21+4 : 31+2 in characteristic two.

Original languageEnglish
Article numbere13
Number of pages20
JournalForum of Mathematics, Sigma
Volume11
DOIs
Publication statusPublished - 1 Mar 2023

Bibliographical note

Open Access via the CUP Agreement
Funding Information:
The first author is grateful to City, University of London for its hospitality during the research for this paper, and to Ehud Meir for conversations about the proof of Theorem . The second author acknowledges support from EPSRC grant EP/T004592/1.

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