Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For ${\mathcal{O}}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantized version of the group algebra of the semidirect product $P\rtimes L$.

中文翻译：

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具有正常Abelian缺陷和Abelian p'惯性商的块

令$ k $为特征$ p $的代数封闭域，令$ {\ mathcal {O}} $为$ k $或其维特环$ W（k）$。假设$ G $为有限组，$ B $为带有正常阿贝尔缺陷组和阿贝尔$ p ^ {\ prime} $惯性商$ L $的$ {\ mathcal {O}} G $块。我们证明$ B $是第二次Frobenius扭曲的同构。这样做的动机是，对Frobenius数进行限制是实现多诺万猜想的关键步骤之一。对于$ {\ mathcal {O}} = k $，我们将$ B $的基本代数明确地描述为关系的颤动。它是半直接乘积$ P \ rtimes L $的群代数的量化版本。