Classifying Groups With a Small Number of Subgroups

2022/01/24 by Alexander Betz, David A. Nash · 2 citations
Mathematics · Engineering · #Finite Group Theory Research #graph theory and CDMA systems #Algebraic Geometry and Number Theory

paper · doi:10.1080/00029890.2022.2010493

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 |G|. 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.

Cited by

• A note on finite groups with few conjugacy classes of subgroups
• Solvability of a group based on its number of subgroups

Related

• Classifying groups with a small number of subgroups
• Groups with at most twelve subgroups
• An approximation for the number of subgroups
• An explicit upper bound on the number of subgroups of a finite group
• Groups with at most 13 nonpower subgroups