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.

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.

scholarly journals On isomorphisms of connected Cayley graphs, III

1998 ◽  
Vol 58 (1) ◽  
pp. 137-145 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Simple Group ◽  
Prime Divisor ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Natural Question ◽  
A Finite Group

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

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.

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.

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.

scholarly journals On the uniform spread of almost simple linear groups

2013 ◽  
Vol 209 ◽  
pp. 35-109 ◽  
Author(s):  
Timothy C. Burness ◽  
Simon Guest
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Nonnegative Integer ◽  
Finite Simple Group ◽  
Linear Groups ◽  
A Finite Group

AbstractLet G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x1,…,xk in G there exists y ∊ C such that G = ‹xi,y› for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) ≥ 2 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = ‹PSLn (q),g› is almost simple, then u(G) ≥ 2 (unless G ≅ S6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

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 $G\cong U_4(2)$.

scholarly journals Element orders in covers of finite simple groups of Lie type

2015 ◽  
Vol 14 (04) ◽  
pp. 1550056 ◽  
Author(s):  
Mariya A. Grechkoseeva
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
A Finite Group ◽  
Groups Of Lie Type

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. We study element orders in proper covers of a finite simple group L of Lie type and prove that such a cover always contains an element whose order differs from the element orders of L provided that L is not L4(q), U3(q), U4(q), U5(2), or 3D4(2).

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