Improve the worst-case complexity of finite group isomorphism

Determine whether the worst-case complexity of the finite Group Isomorphism problem, in the generating-matrix, permutation, or Cayley-table input models, can be improved beyond the currently known quasipolynomial-time bounds.

Background

The paper identifies finite Group Isomorphism as a longstanding computational bottleneck. Although the problem is computable in the standard generating-matrix, permutation, and Cayley-table models, the best general worst-case upper bounds remain of the form |G|{O(log |G|)}, which is quasipolynomial in the group order and exponential in the succinct input size. The authors situate their contribution as progress on an important subclass—2-groups of Frattini class 2—rather than a resolution of the general problem.

References

Despite decades of research from both communities, the worst-case complexity of GpI---whether given by generating matrices, permutations, or explicitly listing out its Cayley (multiplication) table---is not known to be better than $|G|{O(\log |G|)}$.

— Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2  (2610.00874 - Grochow et al., 1 Oct 2026) in Section 1, Introduction