Volume 15 (2019) Article 19 pp. 1-25
The Threshold for Subgroup Profiles to Agree is Logarithmic
Keywords: group isomorphism, profiles, $p$-groups
For primes $p> 2$ and $e> 3$ 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 deterministic polynomial-time isomorphism test.