Based on Cohn and Umansâ€™ group-theoretic method, we embed matrix multiplication into several group algebras, including those of cyclic groups, dihedral groups, special linear groups and Frobenius groups. We prove that SL2(Fp) and PSL2(Fp) can realize the matrix tensor ⟨p, p, p⟩, i.e. it is possible to encode p Ã— p matrix multiplication in the group algebra of such a group. We also find the lower bound for the order of an abelian group realizing ⟨n, n, n⟩ is n3. For Frobenius groups of the form Cq Cp, where p and q are primes, we find that the smallest admissible value of q must be in the range p4/3 ≤ q ≤ p2 − 2p + 3. We also develop an algorithm to find the smallest q for a given prime p.
Worcester Polytechnic Institute
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Li, Zimu, "Fast Matrix Multiplication by Group Algebras" (2018). Masters Theses (All Theses, All Years). 131.
Frobenius group, special linear group, dihedral group, cyclic group, representation theory, group-theoretic method, fast matrix multiplication