Faster Isomorphism Testing of $p$-Groups of Frattini Class 2

The finite group isomorphism problem asks to decide whether two finite groups of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> are isomorphic. Improving the classical <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N^{O(\mathrm{I}\mathrm{o}\mathrm{g}N)}$</tex> -time algorithm for group isomorphism is a long-standing open problem. It is generally regarded that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> groups of class 2 and exponent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> form a bottleneck case for group isomorphism in general. The recent breakthrough by Sun (STOC '23) presents an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N^{O\left((\log N)^{5 / 6}\right)}$</tex> -time algorithm for this group class. In this paper, we improve Sun's algorithm by presenting an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N^{{\tilde{O}}\left((\log {N})^{1^{1 / 2}}\right)}$</tex> -time algorithm for this group class. We also extend our result to the more general <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> -groups of Frattini class 2. Our algorithm is obtained by sharpening the key technical ingredients in Sun's algorithm and building connections with other research topics. One intriguing connection is with the maximal and non-commutative ranks of matrix spaces, which have recently received considerable attention in algebraic complexity and computational invariant theory. Results from the theory of Tensor Isomorphism complexity class (Grochow-Qiao, SIAM J. Comput. '23) are utilized to simplify the algorithm and achieve the extension to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> -groups of Frattini class 2.