We describe certain special consequences of certain elementary methods from group theory for studying the algebraic complexity of matrix multiplication, as developed by H. Cohn, C. Umans et. al. in 2003 and 2005. The measure of complexity here is the exponent of matrix multiplication, a real parameter between 2 and 3, which has been conjectured to be 2. More specifically, a finite group may simultaneously "realize" several independent matrix multiplications via its regular algebra if it has a family of triples of "index" subsets which satisfy the so-called simultaneous triple product property (STPP), in which case the complexity of these several multiplications does not exceed the rank (complexity) of the algebra. This leads to bounds for the exponent in terms of the size of the group and the sizes of its STPP triples, as well as the dimensions of its distinct irreducible representations. Wreath products of Abelian with symmetric groups appear especially important, in this regard, and we give an example of such a group which shows that the exponent is less than 2.84, and could be possibly be as small as 2.02 depending on the number of simultaneous matrix multiplications it realizes.

翻译：本文阐述了H. Cohn、C. Umans等人于2003年和2005年提出的群论基本方法在研究矩阵乘法代数复杂性方面的若干特殊结论。此处的复杂性度量指标是矩阵乘法指数——一个介于2到3之间的实数参数，学界推测其值应为2。具体而言，若有限群具有满足所谓同步三重积性质（STPP）的“指标”子集三元组族，则该群可通过其正则代数同时“实现”多个独立的矩阵乘法运算，此时这些乘法运算的复杂度不超过该代数的秩（复杂度）。由此可导出矩阵乘法指数与群的大小、其STPP三元组的尺寸以及其不同不可约表示的维数之间的约束关系。在这方面，阿贝尔群与对称群的圈积显得尤为重要。我们给出了此类群的一个示例，证明矩阵乘法指数小于2.84，且根据其实现的同步矩阵乘法数量，该指数可能低至2.02。