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

Abstract

For a finite group G, we associate the quantity|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 p n , 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).

A density result on the sum of element orders of a finite group

“…A proof of this result is given in [9] (see Lemma 4.1), where Lazorec also studies the density of a set formed of some algebraic quantities. Also, we recall that the series ∞ i=1 1 pi is divergent, where p i is the ith prime number.…”

The Second Minimum/Maximum Value of the Number of Cyclic Subgroups of Finite -Groups

“…In [1,8,11,15], the authors deduced the second minimum/maximum value of the number of subgroups on the class P. The aim of this paper is to provide some answers to a similar question that can be formulated as: "Which is the second minimum/maximum value of α and what can be said about the minimum/maximum points?" or, equivalently, "Which is the second minimum/maximum value of the number of cyclic subgroups on the class P and what can be said about the minimum/maximum points?".…”