Theory of Computing
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Title : The Threshold for Subgroup Profiles to Agree is Logarithmic
Authors : James B. Wilson
Volume : 15
Number : 19
Pages : 1-25
URL : http://www.theoryofcomputing.org/articles/v015a019
Abstract
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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.