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

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.

References in corpus (1)

• Isomorphism testing of groups of cube-free order

Cited by in corpus (7)

• Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions
• Isomorphism testing of groups of cube-free order
• Polynomial-time isomorphism testing of groups of most finite orders
• A spectral theory for transverse tensor operators
• On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)
• On the Parallel Complexity of Group Isomorphism via Weisfeiler-Leman
• Subgroups of simple groups are as diverse as possible