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

For a finite group $G$ , define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$ , where $o(g)$ denotes the order of $g\in G$ . We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$ , then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

Download Full-text

An answer to a conjecture on the sum of element orders

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500670 ◽

2020 ◽

pp. 2250067

Author(s):

Morteza Baniasad Azad ◽

Behrooz Khosravi ◽

Morteza Jafarpour

Keyword(s):

Finite Group ◽

Lower Bound ◽

Nilpotent Group ◽

Prime Number ◽

Lower Bounds ◽

Finite Groups ◽

Open Problem ◽

Element Orders ◽

Equality Condition ◽

A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

Download Full-text

ON SPECTRUM OF LINEAR GROUPS OVER THE BINARY FIELD AND RECOGNIZABILITY OF L12(2)

International Journal of Algebra and Computation ◽

10.1142/s0218196706002949 ◽

2006 ◽

Vol 16 (02) ◽

pp. 341-349 ◽

Cited By ~ 3

Author(s):

A. R. MOGHADDAMFAR

Keyword(s):

Finite Group ◽

Computer Program ◽

Simple Group ◽

Linear Groups ◽

Element Orders ◽

Binary Field ◽

Set Of Element Orders ◽

A Finite Group

The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable through its spectrum, if for every finite group H, the equality of the spectra ω(H) = ω(G) implies the isomorphism H ≅ G. In this paper, first we try to write a computer program for computing ω(Ln(2)) for any n ≥ 3. Then, we will show that the simple group L12(2) is recognizable through its spectrum.

Download Full-text

Recognition of Finite Simple Linear Groups L4(2k) by Spectrum

Algebra Colloquium ◽

10.1142/s1005386710000441 ◽

2010 ◽

Vol 17 (03) ◽

pp. 469-474

Author(s):

Mingchun Xu

Keyword(s):

Finite Group ◽

Linear Groups ◽

Element Orders ◽

Set Of Element Orders ◽

A Finite Group

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. Grechkoseeva, Shi and Vasilev have proved that the simple linear groups Ln(2k) are recognizable by spectrum for n=2m≥ 16. In this paper we establish the recognizability for the case n=4.

Download Full-text

Sums of element orders in groups of odd order

International Journal of Algebra and Computation ◽

10.1142/s021819672140004x ◽

2021 ◽

pp. 1-15

Author(s):

Marcel Herzog ◽

Patrizia Longobardi ◽

Mercede Maj

Keyword(s):

Finite Group ◽

Positive Integer ◽

Cyclic Group ◽

Element Orders ◽

Maximum Value ◽

Odd Order ◽

A Finite Group

For a finite group [Formula: see text], let [Formula: see text] denote the sum of element orders of [Formula: see text]. If [Formula: see text] is a positive integer let [Formula: see text] be the cyclic group of order [Formula: see text]. It is known that [Formula: see text] is the maximum value of [Formula: see text] on the set of groups of order [Formula: see text], and [Formula: see text] if and only if [Formula: see text] is cyclic of order [Formula: see text]. In this paper, we investigate the second largest value of [Formula: see text] on the set of groups of order [Formula: see text] and the structure of groups [Formula: see text] of order [Formula: see text] with this value of [Formula: see text] when [Formula: see text] is odd.

Download Full-text

A NEW CHARACTERIZATION OF SOME LINEAR GROUPS BY NSE

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500941 ◽

2013 ◽

Vol 13 (02) ◽

pp. 1350094 ◽

Cited By ~ 5

Author(s):

CHANGGUO SHAO ◽

QINHUI JIANG

Keyword(s):

Finite Group ◽

Linear Group ◽

Linear Groups ◽

Element Orders ◽

Set Of Element Orders ◽

A Finite Group

Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set nse (G) ≔ {mk(G) | k ∈ πe(G)}, we call nse (G) the set of numbers of elements with same order. In this paper, we give a new characterization of simple linear group L2(2a) by its order |L2(2a)| and the set nse (L2(2a)), where either 2a - 1 or 2a + 1 is a prime.

Download Full-text

A new characterization of A7 and A8

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2013-0041 ◽

2013 ◽

Vol 21 (3) ◽

pp. 43-50 ◽

Cited By ~ 2

Author(s):

Alireza Khalili Asboei ◽

Syyed Sadegh Salehi ◽

Ali Iranmanesh

Keyword(s):

Finite Group ◽

Element Orders ◽

Set Of Element Orders ◽

A Finite Group

Abstract Let G be a finite group and πe(G) be the set of element orders of G. Let k ∈ πe (G)and mk be the number of elements of order k in G. Set nse(G):={mk|k ∈ πe (G)}. It is proved that An are uniquely determined by nse(An), where n ∈ {4,5,6}. In this paper, we prove that if G is a group such that nse(G)=nse(An) where n ∈ {7,8}, then G ≅ An.

Download Full-text

NSE characterization of some alternating groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500127 ◽

2014 ◽

Vol 14 (02) ◽

pp. 1550012

Author(s):

Neda Ahanjideh ◽

Bahareh Asadian

Keyword(s):

Finite Group ◽

Element Orders ◽

Alternating Groups ◽

Set Of Element Orders ◽

A Finite Group

Let p ≥ 5 be a prime and n ∈ {p, p + 1, p + 2}. Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set nse (G) = {mk(G) : k ∈ πe(G)}. In this paper, we show that if nse (An) = nse (G), p ∈ π(G) and p2 ∤ |G|, then G ≅ An. As a consequence of our result, we show that if nse (An) = nse (G) and |G| = |An|, then G ≅ An.

Download Full-text