The group isomorphism problem determines whether two groups, given by their
Cayley tables, are isomorphic. For groups with order n, an algorithm with
n(logn+O(1)) running time, attributed to Tarjan, was proposed in the
1970s [Mil78]. Despite the extensive study over the past decades, the current
best group isomorphism algorithm has an n(1/4+o(1))logn running time
[Ros13].
The isomorphism testing for p-groups of (nilpotent) class 2 and exponent
p has been identified as a major barrier to obtaining an no(logn) time
algorithm for the group isomorphism problem. Although the p-groups of class 2
and exponent p have much simpler algebraic structures than general groups,
the best-known isomorphism testing algorithm for this group class also has an
nO(logn) running time.
In this paper, we present an isomorphism testing algorithm for p-groups of
class 2 and exponent p with running time nO((logn)5/6) for any
prime p>2. Our result is based on a novel reduction to the skew-symmetric
matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop
several tools for matrix space analysis, including a matrix space
individualization-refinement method and a characterization of the low rank
matrix spaces.Comment: Accepted to STOC 202