scholarly journals A new characterization of A7 and A8

2013 ◽  
Vol 21 (3) ◽  
pp. 43-50 ◽  
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.

A NEW CHARACTERIZATION OF SOME LINEAR GROUPS BY NSE

2013 ◽  
Vol 13 (02) ◽  
pp. 1350094 ◽  
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.

NSE characterization of some alternating groups

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.

A NEW CHARACTERIZATION OF PGL(2, p)

2013 ◽  
Vol 12 (07) ◽  
pp. 1350040 ◽  
Author(s):  
ALIREZA KHALILI ASBOEI
Keyword(s):  
Finite Group ◽  
Prime Divisor ◽  
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. Let k ∈ πe(G) and mk be the number of elements of order k in G. Set nse (G) ≔ {mk ∣ k ∈ πe(G)}. In this paper, it is proved if G is a group with the following properties, then G ≅ PGL (2, p). (1) p > 3 is prime divisor of ∣G∣ but p2 does not divide ∣G∣. (2) nse (G) = nse ( PGL (2, p)).

scholarly journals A CHARACTERIZATION OF U4(2) BY NSE

2019 ◽  
pp. 641
Author(s):  
Farnoosh Hajati ◽  
Ali Iranmanesh ◽  
Abolfazl Tehranian
Keyword(s):  
Finite Group ◽  
Element Orders ◽  
Set Of Element Orders ◽  
A Finite Group

‎Let $G$ be a finite group and $\omega(G)$ be the set of element orders of $G$‎. ‎Let $k\in\omega(G)$ and $m_k$ be the number of elements of order $k$ in $G$‎. ‎Let $ nse(G)=\{m_k|k\in \omega(G)\}$‎. ‎The aim of this paper is to prove that‎, ‎if $G$ is a finite group such that nse($G$)=nse($U_4(2)$)‎, ‎then $G\cong U_4(2)$.

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

2006 ◽  
Vol 16 (02) ◽  
pp. 341-349 ◽  
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.

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

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.

RECOGNITION OF SOME FINITE SIMPLE GROUPS OF TYPE Dn(q) BY SPECTRUM

2009 ◽  
Vol 19 (05) ◽  
pp. 681-698 ◽  
Author(s):  
HUAIYU HE ◽  
WUJIE SHI
Keyword(s):  
Finite Group ◽  
Finite Simple Group ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
Set Of Element Orders ◽  
A Finite Group ◽  
Group D ◽  
Nonabelian Composition Factor

The spectrum ω(G) of a finite group G is the set of element orders of G. Let L be finite simple group Dn(q) with disconnected Gruenberg–Kegel graph. First, we establish that L is quasi-recognizable by spectrum except D4(2) and D4(3), i.e., every finite group G with ω(G) = ω(L) has a unique nonabelian composition factor that is isomorphic to L. Second, for some special series of integers n, we prove that L is recognizable by spectrum, i.e., every finite group G with ω(G) = ω(L) is isomorphic to L.

A New Characterization of PSL(2, q) by Group Order and the Set of Vanishing Element Orders

2019 ◽  
Vol 26 (03) ◽  
pp. 459-466
Author(s):  
Changguo Shao ◽  
Qinhui Jiang
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Complex Character ◽  
Element Orders ◽  
Group Order ◽  
Vanishing Elements ◽  
A Finite Group

An element g in a finite group G is called a vanishing element if there exists some irreducible complex character χ of G such that [Formula: see text]. Denote by Vo(G) the set of orders of vanishing elements of G, and we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime power.

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

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