We show that the Green correspondence induces an injective group homomorphism from the linear source Picard group $\mathcal{L}(B)$ of a block $B$ of a finite group algebra to the linear source Picard group $\mathcal{L}(C)$, where $C$ is the Brauer correspondent of $B$. This homomorphism maps the trivial source Picard group $\mathcal{T}(B)$ to the trivial source Picard group $\mathcal{T}(C)$. We show further that the endopermutation source Picard group $\mathcal{E}(B)$ is bounded in terms of the defect groups of $B$ and that when $B$ has a normal defect group $\mathcal{E}(B)=\mathcal{L}(B)$. Finally we prove that the rank of any invertible $B$-bimodule is bounded by that of $B$.

中文翻译：

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线性源可逆双模和格林对应

我们表明，格林对应从有限群代数块 $B$ 的线性源 Picard 群 $\mathcal{L}(B)$ 到线性源 Picard 群 $\mathcal{L} (C)$，其中 $C$ 是 $B$ 的 Brauer 通讯员。这种同态将平凡源 Picard 群 $\mathcal{T}(B)$ 映射到平凡源 Picard 群 $\mathcal{T}(C)$。我们进一步表明，endopermutation 源 Picard 群 $\mathcal{E}(B)$ 以 $B$ 的缺陷组为界，并且当 $B$ 具有正常缺陷组 $\mathcal{E}(B )=\mathcal{L}(B)$。最后我们证明任何可逆的$B$-bimodule 的秩都受$B$ 的秩的限制。