On the maximum number of subgroups of a finite group

2023/04/27 by Fusari, Marco, Spiga, Pablo
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2304.14200

Abstract

Given a finite group R, we let Sub(R) denote the collection of all subgroups of R. We show that |Sub(R)|< c⋅ |R|^{(log₂|R|)/(4)}, where c<7.372 is an explicit absolute constant. This result is asymptotically best possible. Indeed, as |R| tends to infinity and R is an elementary abelian 2-group, the ratio \frac|Sub(R)||R|^{(log₂|R|)/(4)} tends to c.

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