Theory of Computing
-------------------
Title     : Groups with Identical $k$-Profiles
Authors   : George Glauberman and Lukasz Grabowski
Volume    : 11
Number    : 15
Pages     : 395-401
URL       : https://theoryofcomputing.org/articles/v011a015

Abstract
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We show that for $1 \le k \le \sqrt{2\log_3 n}-(5/2)$, the multiset of
isomorphism types of $k$-generated subgroups does not determine a
group of order at most $n$. This answers a question raised by Tim
Gowers in connection with the Group Isomorphism problem.