OD-Characterization of the Symmetric Group S49

2012 ◽  
Vol 535-537 ◽  
pp. 2596-2599 ◽  
Author(s):  
Yan Xiong Yan
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Degree Pattern ◽  
A Finite Group

The degree pattern of a finite group G associated with its prime graph has been introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |G|=|H| and D(G)=D(H).Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In this paper, we will show that the symmetric group S49 can be characterized by its order and degree pattern. In fact, the symmetric group S49 is 3-fold OD-characterizable

scholarly journals On alternating and symmetric groups which are quasi OD-characterizable

2017 ◽  
Vol 16 (04) ◽  
pp. 1750065 ◽  
Author(s):  
Ali Reza Moghaddamfar
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Symmetric Groups ◽  
Degree Pattern ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

OD-Characterization of Symmetric Group S57

2013 ◽  
Vol 834-836 ◽  
pp. 1799-1802
Author(s):  
Mei Yang
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Degree Pattern ◽  
A Finite Group

In this paper, we show that the symmetric group can be characterized by its order and degree pattern. In fact, we get the following theorem: Let G be a finite group such that and . Then G is isomorphisic to one of the almost simple groups: and . Particularly, is 3-fold OD-characterizable.

OD-Characterization of Certain Finite Groups Having Connected Prime Graphs

2010 ◽  
Vol 17 (01) ◽  
pp. 121-130 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Alternating Group ◽  
Degree Pattern ◽  
Prime Graphs ◽  
Group A ◽  
A Finite Group

The degree pattern of a finite group G denoted by D(G) was introduced in [5]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and same degree pattern as G. In the present article, we show that the alternating group A10 and the automorphism group Aut (McL) are 2-fold OD-characterizable, while the automorphism group Aut (J2) is 3-fold OD-characterizable and the symmetric group S10 is 8-fold OD-characterizable. It is worth mentioning that the prime graphs associated to these groups are connected and, in fact, among the groups with this property, they are the first groups which are investigated for OD-characterizability.

A NEW CHARACTERIZATION OF ALMOST SPORADIC GROUPS

2002 ◽  
Vol 01 (03) ◽  
pp. 267-279 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Sporadic Groups ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Order Components ◽  
Positive Integers ◽  
Coprime Positive Integers

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut (J2) and Aut (McL), and the automorphism group of PSL(2, 2n) where n=2sare also uniquely determined by their order components. Also we discuss about the characterizability of Aut (PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

2005 ◽  
Vol 12 (03) ◽  
pp. 431-442 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Natural Numbers ◽  
Prime Graphs ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pkand natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Apwith p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

A Characterization of Some Finite Simple Groups Through Their Orders and Degree Patterns

2012 ◽  
Vol 19 (03) ◽  
pp. 473-482 ◽  
Author(s):  
M. Akbari ◽  
A. R. Moghaddamfar ◽  
S. Rahbariyan
Keyword(s):  
Finite Group ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Degree Pattern ◽  
A Finite Group ◽  
Mersenne Prime

The degree pattern of a finite group G was introduced in [15] and denoted by D (G). A finite group M is said to be OD-characterizable if G ≅ M for every finite group G such that |G|=|M| and D (G)= D (M). In this article, we show that the linear groups Lp(2) and Lp+1(2) are OD-characterizable, where 2p-1 is a Mersenne prime. For example, the linear groups L2(2) ≅ S3, L3(2) ≅ L2(7), L4(2) ≅ A8, L5(2), L6(2), L7(2), L8(2), L13(2), L14(2), L17(2), L18(2), L19(2), L20(2), L31(2), L32(2), L61(2), L62(2), L89(2), L90(2), etc., are OD-characterizable. We also show that the simple groups L4(5), L4(7) and U4(7) are OD-characterizable.

OD-Characterization of Simple K4-Groups

2009 ◽  
Vol 16 (02) ◽  
pp. 275-282 ◽  
Author(s):  
Liangcai Zhang ◽  
Wujie Shi
Keyword(s):  
Finite Group ◽  
Degree Pattern ◽  
A Finite Group

Let M ≠ A10be a simple K4-group. In this paper, we prove that if G is a finite group with the same order and degree pattern as M, then G ≅ M. Equivalently, all simple K4-groups except A10are OD-characterizable. Moreover, we put forward an example to show that A10and J2× Z3have the same order and degree pattern, but they are not isomorphic.

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.

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

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