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).

scholarly journals A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

Quasirecognition of E6(q) by the orders of maximal abelian subgroups

2018 ◽  
Vol 17 (07) ◽  
pp. 1850122 ◽  
Author(s):  
Zahra Momen ◽  
Behrooz Khosravi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Simple Group ◽  
Outer Automorphism ◽  
Composition Factor ◽  
Outer Automorphism Group ◽  
Abelian Subgroups ◽  
A Finite Group ◽  
Nonabelian Composition Factor

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

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 Simple Group U3(5)

1970 ◽  
Vol 38 ◽  
pp. 27-40 ◽  
Author(s):  
Koichiro Harada
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Permutation Group ◽  
Transitive Permutation Group ◽  
List Type ◽  
A Finite Group

In this note we consider a finite group G which satisfies the following conditions: (0. 1) G is a doubly transitive permutation group on a set Ω of m + 1 letters, where m is an odd integer ≥ 3,(0. 2) if H is a subgroup of G and contains all the elements of G which fix two different letters α, β, then H contains unique permutation h0 ≠ 1 which fixes at least three letters,(0. 3) every involution of G fixes at least three letters,(0. 4) G is not isomorphic to one of the groups of Ree type.

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.

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.

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

scholarly journals A characterization of finite p-soluble groups of p-length one by commutator identities

1981 ◽  
Vol 30 (3) ◽  
pp. 257-263 ◽  
Author(s):  
Rolf Brandl
Keyword(s):  
Finite Group ◽  
Classical Result ◽  
Soluble Groups ◽  
Similar Description ◽  
A Finite Group ◽  
Engel Condition ◽  
Almost All

AbstractA classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

scholarly journals On the nilpotency of the solvable radical of a finite group isospectral to a simple group

2020 ◽  
Vol 23 (3) ◽  
pp. 447-470
Author(s):  
Nanying Yang ◽  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil’ev
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Solvable Radical ◽  
A Finite Group ◽  
Orders Of Elements

AbstractWe refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that, except for one specific case, the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

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