A connection between the number of subgroups and the order of a finite group

2019/01/18 by Lazorec, Mihai-Silviu · 2 citations
#20D60 #20K01 #FOS: Mathematics #Group Theory (math.GR) #Primary 20D30 #Secondary 20D15

paper · doi:10.48550/arxiv.1901.06425

Abstract

For a finite group G, we associate the quantity β(G)=(|L(G)|)/(|G|), where L(G) is the subgroup lattice of G. Different properties and problems related to this ratio are studied throughout the paper. We determine the second minimum value of β on the class of p-groups of order pⁿ, where n≥ 3 is an integer. We show that the set containing the quantities β(G), where G is a finite (abelian) group, is dense in [0,∞). Finally, we consider β to be a function on L(G) and we mark some of its properties, the main result being the classification of finite abelian p-groups G satisfying β(H)≤ 1, ∀ H∈ L(G).

Cited by

• A note on connecting multiplicative functions defined on finite groups
• Finite groups with mostly involuted cyclic subgroups

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