Breaking the nlog n Barrier for Solvable-Group Isomorphism

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G ∼= H. The n barrier for group isomorphism has withstood all attacks — even for the special cases of p-groups and solvable groups — ever since the n generator-enumeration algorithm. Following a framework due to Wagner, we present the first significant improvement over n by reducing group isomorphism to composition-series isomorphism which is then reduced to low-degree graph isomorphism. We show that group isomorphism is n logp n+O(1) Turing reducible to composition-series isomorphism where p is the smallest prime dividing the order of the group. Combining our reduction with an n algorithm for p-group composition-series isomorphism, we obtain an n logn+O(1) algorithm for p-group isomorphism. We then generalize our techniques from p-groups using Sylow bases to derive an n logn+O(logn/ log logn) algorithm for solvable-group isomorphism. Finally, we relate group isomorphism to the collision problem which allows us replace the 1/2 in the exponents with 1/4 using randomized algorithms and 1/6 using quantum algorithms. ar X iv :1 20 5. 06 42 v6 [ cs .D S] 1 1 D ec 2 01 3