Detecting structural properties of finite groups by the sum of element orders
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DETECTING STRUCTURAL PROPERTIES OF FINITE GROUPS BY THE SUM OF ELEMENT ORDERS BY
˘rna ˘uceanu Marius Ta Faculty of Mathematics, “Al. I. Cuza” University, Ia¸si, Romania e-mail: [email protected]
ABSTRACT
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
1. Introduction Given a finite group G, we consider the function ψ(G) = o(x), x∈G
where o(x) denotes the order of x. This has been introduced by H. Amiri, S. M. Jafarian Amiri and I. M. Isaacs [1]. They proved the following theorem: Theorem A: If G is a group of order n, then ψ(G) ≤ ψ(Cn ), and we have equality if and only if G is cyclic. In other words, the cyclic group Cn is the unique group of order n which attains the maximal value of ψ(G) among groups of order n. Since then many authors have studied the function ψ(G) and its relations with the structure of G (see, e.g., [2], [3], [5], [6], [7], [8], [9], [12], [14]). In the papers [4] and [12] M. Amiri and S. M. Jafarian Amiri, and, independently, R. Shen, G. Chen and C. Wu started the investigation of groups with the second Received May 9, 2019 and in revised form July 21, 2019
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˘ ˘ M. TARN AUCEANU
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largest value of the sum of element orders. M. Herzog, P. Longobardi and M. Maj [6] determined the exact upper bound for ψ(G) for non-cyclic groups of order n: Theorem B: If G is a non-cyclic group of order n and q is the least prime divisor of the order of n, then ψ(G) ≤
[(q 2 − 1)q + 1](q + 1) ψ(Cn ) = f (q)ψ(Cn ). q5 + 1
Moreover, the equality holds if and only if n = q 2 m with (m, q!) = 1 and G∼ = (Cq × Cq ) × Cm . Note that the above function f is strictly decreasing on [2, ∞). Consequently, we have 7 ψ(Cn ), ψ(G) ≤ f (2)ψ(Cn ) = 11 and the equality holds for n = 4m with m odd and G ∼ = (C2 × C2 ) × Cm . By using the sum of element orders, several criteria for solvability of finite groups have been determined (see, e.g., [5, 7]). We recall here the following theorem of M. Baniasad Asad and B. Khosravi [5]: Theorem C: If G is a group of order n and ψ(G) > solvable.
211 1617
ψ(Cn ), then G is
Note that the equality 211 ψ(Cn ) 1617 occurs for n = 60m with (30, m) = 1 and G ∼ = A5 × Cm . We also recall a criterion for nilpotency of finite groups that has been proved in [14]: ψ(G) =
Theorem D: If G is a group of order n and ψ(G) > 13 21 ψ(Cn ), then G is 13 nilpotent. Moreover, we have ψ(G) = 21 ψ(Cn ) if and only if n = 6m with (6, m) = 1 and G ∼ = S3 × Cm . The four largest values of the ratio ψ (G) =
ψ(G) ψ(C|G| )
and the groups G for which they are attained can be obtained from Theorem D.
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STRUCTURAL PROPERTIES OF FINITE GROUPS
Corollary E: Let G be a finite group satisfying ψ (G) > 27 7 , ,1 , ψ (G) ∈ 43 11 and one of the following holds: (a) G ∼ = Q8 × Cm , where m is odd; ∼ (C2 × C2 ) × Cm , where m is odd; (b) G =
13 21 .
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Then
(c) G is cyclic. The above results show that a
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