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 -------- 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.