Volume 11 (2015) Article 15 pp. 395-401 [Note]
Groups with Identical $k$-Profiles
Keywords: group theory, nilpotent groups, $p$-groups, group isomorphism problem, algorithms, lower bounds, $k$-generated group, $k$-profile of groups
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.