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Volume 11 (2015) Article 15 pp. 395-401 [Note]
Groups with Identical $k$-Profiles
Received: June 16, 2013
Revised: November 30, 2015
Published: December 23, 2015
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Keywords: group theory, nilpotent groups, $p$-groups, group isomorphism problem, algorithms, lower bounds, $k$-generated group, $k$-profile of groups
ACM Classification: F.2.2
AMS Classification: 68Q17, 20D15, 20F69, 68Q25

Abstract: [Plain Text Version]

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.