## 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 p^{3} 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 2^{1}+^{4} : 3^{1}+^{2} in characteristic two.

Original language | English |
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Article number | e13 |

Number of pages | 20 |

Journal | Forum of Mathematics, Sigma |

Volume | 11 |

DOIs | |

Publication status | Published - 1 Mar 2023 |

### Bibliographical note

Open Access via the CUP AgreementFunding 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.