Classifying Groups With a Small Number of Subgroups

Abstract

We provide lower bounds on the number of subgroups of a group G as a function of the primes and exponents appearing in the prime factorization of $\left|G\right|$. Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and all nonabelian groups with 19 or fewer subgroups. This allows us to extend the integer sequence A274847 in the On-Line Encyclopedia of Integer Sequences introduced by Slattery.

MSC:

• Primary 20E07
• Secondary 20D20

Acknowledgment

The authors wish to thank the editor and referees for helpful suggestions.

Notes

¹ Groups, Algorithms, and Programming—see www.gap-system.org.

² Note: The statement of Theorem 4.2.1 in [1] has a typo, but the proof proves the statement given here and [16] confirms this result.

³ We thank Aivazidis for bringing this to our attention.

⁴ Note that this situation exactly corresponds to G having only normal Sylow subgroups.

Additional information

Notes on contributors

Alexander Betz

Alexander Betz is a currently a graduate student at Stony Brook University. He graduated from Le Moyne College with a Bachelor of Arts in Mathematics in May 2020. He has worked on three different papers in finite group theory and has enjoyed the field thoroughly. As a graduate student he is excited to learn more about all facets of mathematics and find a field of math to dedicate himself to. Le Moyne College, Syracuse, NY 13214 [email protected]

David A. Nash

David A. Nash is an associate professor of mathematics at Le Moyne College. After an undergraduate career at Santa Clara University, he earned his Ph.D. from University of Oregon in 2010 with an emphasis in representation theory. He enjoys sharing his passion for mathematics and problem solving with his students. He has especially appreciated involving undergraduates in research over the past several years by tackling interesting problems with more recreational origins. Department of Mathematics, Statistics, and Computer Science, Le Moyne College, Syracuse, NY 13214 [email protected]