The threshold for subgroup profiles to agree is $\Omega(\log n)$

TL;DR

This paper constructs specific finite groups demonstrating that subgroup profile agreement requires at least logarithmic size, challenging assumptions about efficient isomorphism testing for all finite groups.

Contribution

It provides explicit examples of groups with identical subgroup and quotient structures, showing a lower bound for subgroup profile agreement necessary for isomorphism tests.

Findings

Groups with identical subgroup and quotient multisets exist for large primes

Subgroup profile agreement threshold is at least logarithmic in group size

Constructs a class of groups with polylogarithmic-time isomorphism testing

Abstract

For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent speculation concerning a path towards efficient isomorphism tests for general finite groups. These groups have a special purpose polylogarithmic-time isomorphism test.