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The Cohn-Umans group-theoretic approach to matrix multiplication suggestsembedding matrix multiplication into group algebra multiplication, and bounding$\omega$ in terms of the representation theory of the host group. Thisframework is general enough to capture the best known upper bounds on $\omega$and is conjectured to be powerful enough to prove $\omega = 2$, althoughfinding a suitable group and constructing such an embedding has remainedelusive. Recently it was shown, by a generalization of the proof of the Cap SetConjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$in this framework, which ruled out a family of potential constructions in theliterature.  In this paper we study nonabelian groups as potential hosts for an embedding.We prove two main results:  (1) We show that a large class of nonabelian groups---nilpotent groups ofbounded exponent satisfying a mild additional condition---cannot prove $\omega= 2$ in this framework. We do this by showing that the shrinkage rate of powersof the augmentation ideal is similar to the shrinkage rate of the number offunctions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials;our proof technique can be seen as a generalization of the polynomial methodused to resolve the Cap Set Conjecture.  (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on$\omega$ when the embedding is via three Young subgroups---subgroups of theform $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is anatural strategy that includes all known constructions in $S_n$.  By developing techniques for negative results in this paper, we hope tocatalyze a fruitful interplay between the search for constructions provingbounds on $\omega$ and methods for ruling them out.

Which groups are amenable to proving exponent two for matrix multiplication?