By an {\em isotropy group} of a tensor $t\in V_1 \otimes V_2\otimes V_3=\widetilde V$ we mean the group of all invertible linear transformations of $\widetilde V$ that leave $t$ invariant and are compatible (in an obvious sense) with the structure of tensor product on~$\widetilde V$. We consider the case where $t$ is the structure tensor of multiplication map of rectangular matrices. The isotropy group of this tensor was studied in 1970s by de Groote, Strassen, and Brockett-Dobkin. In the present work we enlarge, make more precise, expose in the language of group actions on tensor spaces, and endow with proofs the results previously known. This is necessary for studying the algorithms of fast matrix multiplication admitting symmetries. The latter seems to be a promising new way for constructing fast algorithms.

翻译：V_1 otimes V_2\ otimes V_3 {bloblytilde V$,我们指的是所有无法倒置的线性变换 $@Unitilde V$ 的组合,它留下了一美元,并且(在明显意义上)与 $@unor 产品结构在~$\ loytilde V$上兼容。我们认为美元是矩形矩阵乘法结构的振幅。 1970年代, de Groote、 Strassen 和 Brockett- Dobkin 研究了这个变异的组合。 在目前的工作中,我们扩大、 更精确地展示了在 Exmor 空间的群集行动语言, 并用先前已知的结果来证明。 这是研究快速矩阵乘法组合的算法所必要的。 后者似乎是构建快速算法的有希望的新方法 。