Breaking the n^(log 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 and H are isomorphic. The n^(log n) barrier for group isomorphism has withstood all attacks — even for the special cases of p-groups and solvable groups — ever since the n^(log n + O(1)) generator-enumeration algorithm. In this work, we present the first significant improvement over n^(log n) by showing that group isomorphism is n^((1 / 2) log_p 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^(O(p / log p)) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log n + O(1)) algorithm for p-group isomorphism. We then generalize our techniques from p-groups using Sylow bases to derive an n^((1 / 2) log n + O(log n / log log n)) 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.

💡 Research Summary

The paper tackles the classic group isomorphism problem—given two finite groups by their multiplication tables, decide whether they are isomorphic—and breaks the long‑standing n^(log n) time barrier that has persisted even for restricted families such as p‑groups and solvable groups. The authors achieve this by a sequence of reductions and algorithmic innovations that together lower the exponent from log n to roughly (1/2)·log n, and further improve it with randomized and quantum techniques.

The first major contribution is a reduction from general group isomorphism to composition‑series isomorphism. A composition series is a chain of normal subgroups whose successive quotients are simple. By focusing on the smallest prime p dividing the group order, the authors show that one can isolate the p‑group part of each composition factor and treat it separately. They devise an algorithm for p‑group composition‑series isomorphism that runs in n^(O(p/ log p)) time, which is dramatically faster than the naïve n^(log n) approach because the number of possible series is bounded by a function of p rather than the whole group size. Consequently, the reduction incurs only a factor of n^((1/2)·log_p n+O(1)), yielding an overall n^((1/2)·log n+O(1)) algorithm for p‑group isomorphism.

The second contribution extends this technique to all solvable groups. Solvable groups admit a Sylow basis: a collection of Sylow p‑subgroups that intersect trivially and generate the whole group. The authors construct a “Sylow‑basis‑based composition series” by independently building composition series for each Sylow p‑subgroup and then weaving them together. The combinatorial explosion that normally appears when mixing different primes is controlled by a careful analysis that shows the extra overhead grows only as O(log n/ log log n). This leads to a deterministic algorithm for solvable‑group isomorphism with running time n^((1/2)·log n+O(log n/ log log n)).

Finally, the paper connects group isomorphism to the collision problem, a well‑studied task in both classical randomized and quantum computing. By interpreting the search for a suitable isomorphism as a collision‑finding problem, the authors replace the deterministic exhaustive search with algorithms that find collisions in roughly the square root of the search space. This reduces the exponent from 1/2 to 1/4 for randomized algorithms. Moreover, by employing quantum collision‑finding (e.g., using Shor’s algorithm or quantum walks), the exponent can be lowered further to 1/6.

In summary, the authors present a four‑step breakthrough: (1) reduce group isomorphism to composition‑series isomorphism; (2) solve p‑group composition‑series isomorphism in n^(O(p/ log p)) time; (3) lift the result to all solvable groups via a Sylow‑basis construction; and (4) accelerate the final search using classical randomization and quantum techniques. The deterministic bound of n^((1/2)·log n+polylog n) shatters the historic n^(log n) barrier, and the randomized/quantum refinements push the exponent down to 1/4 and 1/6 respectively. This work not only advances the theoretical understanding of group isomorphism complexity but also opens new avenues for algorithmic group theory, suggesting that similar reductions and collision‑based strategies might eventually break the barrier for even broader classes of groups.