On the Simple Group of J. Tits

1973 ◽  
Vol 16 (1) ◽  
pp. 87-92
Author(s):  
David Parrott
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Simple Subgroup ◽  
Index 2

In the series of simple groups 2F4(q),q =2 2m+1, discovered by Ree, Tits [4] showed that the group 2F4(2) was not simple but contained a simple subgroup of index 2. In this note we extend the characterization of obtained by the author in [3].

scholarly journals A Characterization of the Finite Simple Groups PSp(4, q), G2(q), I

1969 ◽  
Vol 36 ◽  
pp. 143-184 ◽  
Author(s):  
Paul Fong ◽  
W.J. Wong
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Symplectic Group ◽  
Prime Power ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Central Product ◽  
Twisted Group ◽  
Index 2

Suppose that G is the projective symplectic group PSp(4, q), the Dickson group G2(q)> or the Steinberg “triality-twisted” group where q is an odd prime power. Then G is a finite simple group, and G contains an involution j such that the centralizer C(j) in G has a subgroup of index 2 which contains j and which is the central product of two groups isomorphic with SL(2,q1) and SL(2,q2) for suitable ql q2. We wish to show that conversely the only finite simple groups containing an involution with this property are the groups PSp(4,q), G2(q)9. In this first paper we shall prove the following result.

A Characterization of the Tits' Simple Group

1972 ◽  
Vol 24 (4) ◽  
pp. 672-685 ◽  
Author(s):  
David Parrott
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Frobenius Group ◽  
Weakly Closed ◽  
Even Order ◽  
Simple Subgroup ◽  
Index 2 ◽  
Ree Group ◽  
A Finite Group

In [6], J. Tits has shown that the Ree group 2F4(2) is not simple but possesses a simple subgroup of index 2. In this paper we prove the following theorem:THEOREM. Let G be a finite group of even order and let z be an involution contained in G. Suppose H = CG(z) has the following properties:(i) J = O2(H) has order 29and is of class at least 3.(ii) H/J is isomorphic to the Frobenius group of order 20.(iii) If P is a Sylow 5-subgroup of H, then Cj(P) ⊆ Z(J).Then G = H • O(G) or G ≊ , the simple group of Tits, as defined in [6].For the remainder of the paper, G will denote a finite group which satisfies the hypotheses of the theorem as well as G ≠ H • O(G). Thus Glauberman's theorem [1] can be applied to G and we have that 〈z〉 is not weakly closed in H (with respect to G). The other notation is standard (see [2], for example).

On the Mathieu group M23

1971 ◽  
Vol 12 (4) ◽  
pp. 385-392 ◽  
Author(s):  
N. Bryce
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Unique Determination ◽  
Mathieu Group ◽  
Mathieu Groups

Until 1965, when Janko [7] established the existence of his finite simple group J1, the five Mathieu groups were the only known examples of isolated finite simple groups. In 1951, R. G. Stanton [10] showed that M12 and M24 were determined uniquely by their order. Recent characterizations of M22 and M23 by Janko [8], M22 by D. Held [6], and M11 by W. J. Wong [12], have facilitated the unique determination of the three remaining Mathieu groups by their orders. D. Parrott [9] has so characterized M22 and M11, while this paper is an outline of the characterization of M23 in terms of its order.

A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH

2009 ◽  
Vol 08 (01) ◽  
pp. 105-114 ◽  
Author(s):  
LIANGCAI ZHANG ◽  
WUJIE SHI
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Nonabelian Simple Group ◽  
Prime Graphs ◽  
Vertex Set ◽  
Thompson’S Conjecture ◽  
A Finite Group ◽  
Noncommuting Graph ◽  
Almost All

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

scholarly journals Characterization of a family of simple groups by their character table

1977 ◽  
Vol 24 (3) ◽  
pp. 296-304 ◽  
Author(s):  
Marcel Herzog ◽  
David Wright
Keyword(s):  
Conjugacy Class ◽  
Simple Group ◽  
Character Table ◽  
Simple Groups

AbstractThe paper establishes a method for bounding the 2-rank of a simple group with one conjugacy class of involutions, by means of its character table. For many groups of 2-rank ≦ 4, this bound is shown to be exact. The main result is that the simple groups G2(q),(q,6) = 1, are characterized bv their character table.

scholarly journals Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

2016 ◽  
Author(s):  
Michel Planat ◽  
Hishamuddin Zainuddin
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Physical Significance ◽  
Simple Groups ◽  
Finite Geometries ◽  
Group Representations ◽  
Small Index ◽  
Point Line

Every finite simple group P can be generated by two of its elements. Pairs of generators forP are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual but significant to recognize that a P is a Grothendieck’s dessin d’enfant D and that a wealth of standard graphs and finite geometries G - such as near polygons and their generalizations - are stabilized by a D. In our paper, tripods P − D − G of rank larger than two,corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurationsdefined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G's have a contextuality parameterclose to its maximal value 1.

A Characterization of PSU11(q)

2004 ◽  
Vol 47 (4) ◽  
pp. 530-539 ◽  
Author(s):  
A. Iranmanesh ◽  
B. Khosravi
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Order Components

AbstractOrder components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups PSU11(q) are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on PSU11(q) are obtained.

scholarly journals A new characterization of the simple groups C4(q), by its order and the largest order of elements

2020 ◽  
Vol 23 (2) ◽  
pp. 283-290
Author(s):  
Behnam Ebrahimzadeh ◽  
Reza Mohammadyari ◽  
Miryousef Sadeghi
Keyword(s):  
Simple Group ◽  
Prime Numbers ◽  
Simple Groups ◽  
Order Of Elements

We prove that the simple group C4(q), where q > 2 and (q4 + 1)=2 are prime numbers, can be uniquely determined by its order and the largest order of elements.

A CHARACTERIZATION OF SPORADIC SIMPLE GROUPS BY NSE AND ORDER

2012 ◽  
Vol 12 (02) ◽  
pp. 1250158 ◽  
Author(s):  
ALIREZA KHALILI ASBOEI ◽  
SEYED SADEGH SALEHI AMIRI ◽  
ALI IRANMANESH ◽  
ABOLFAZL TEHRANIAN
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Sporadic Simple Groups ◽  
A Finite Group

Let G be a finite group and nse (G) the set of numbers of elements with the same order in G. In this paper, we prove that if ∣G∣ = ∣S∣ and nse (G) = nse (S), where S is a sporadic simple group, then the finite group G is isomorphic to S.

A Note on the Adjacency Criterion for the Prime Graph and the Characterization of Cp(3)

2012 ◽  
Vol 19 (03) ◽  
pp. 553-562 ◽  
Author(s):  
Huaiyu He ◽  
Wujie Shi
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Prime Graph ◽  
Recognition Problem ◽  
Simple Groups ◽  
Finite Simple Groups

In this paper, we remedy some errors in the paper [17]; in particular, for any non-abelian finite simple group G other than Alt n such that n-2, n-1 and n are not primes, we prove that for each r ∈ π(G), there always exists s ∈ π(G) which is non-adjacent to r in the Gruenberg-Kegel graph of G. Applications of these results to the recognition problem of some finite simple groups by spectrum are also considered.

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