An Investigation on the Parabolic Subgroups of the General Linear Groups by Using GAP

A typical example for the algebraic groups is the general linear groups G=GL(n,F), we have studied the structure of such groups and paid special attention to its important substructures, namely the Parabolic subgroups. For a given G we computed all the Parabolic subgroups and determined their number, depending on the fact that any finite group has a composition series and the composition factors of a composition series are simple groups which are completely classified, we report here some investigations on the computed Parabolic subgroups. This has been done with the utility of GAP.

Abelian quotients and orbit sizes of linear groups

Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.

Regular and p-Regular Orbits of Solvable Linear Groups,Ⅱ

Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.

Uniform Boundedness of Torsion Subgroups of Linear Groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ（2m ＋ 1, F） over a linearable field F of characteristic 0 are shown to be complete groups, provided m 〉 3.

A RECOGNITION OF SIMPLE GROUPS PSL(3,q) BY THEIR ELEMENT ORDERS

For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.

Two Classes of Solvable Subgroups of SL（3, C） and Application for Differential Equations in C

Structures of two classes of solvable subgroups in SL（3, C） are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is discussed.

Extended Group of the Subgroup Gr in Linear Group Over the Principal Ideal Ring

Suppose R is a principal ideal ring, R^＊ is a multiplicative group which is composed of all reversible elements in R, and Mn（R）, GL（n,R）, SL（n,R） are denoted by,Mn（R）={A=（aij）n×n｜aij∈R,i,j=1,2…,n},GL（n,R）={g｜g∈Mn（R）,det g∈R^＊},SL（n,R）={g∈GL（n,R）｜det g=1},SL（n,R）≤G≤GL（n,R）（n≥3）,respectively,then basing on these facts, this paper mainly focus on discussing all extended groups of Gr={（O D^A B）∈G｜A∈GL（r,R）,（1≤r〈n）}in G when R is a principal ideal ring.

THE AUTOMORPHISM GROUP OF A LINEAR CODE

We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-

On the Structure of Some Groups Containing L2 （11） wrM12

In this paper, we generate the wreath product L2 （1 1） wrM12 using only two permutations. Also, we show the structure of some groups containing the wreath product L2（1 1）wrM12. The structure of the groups founded is determined in terms of wreath product （L2 （11）wrM12）wrCt. Some related cases are also included. Also, we will show that S132K＋1 and A132K＋l can be generated using the wreath product （L2 （1 1）wrM12） wr Ck and a transposition in S132K＋1 and an element of order 3 in A132K＋l. We will also show that S132K＋1 and A132K＋1 can be generated using the wreath product L2 （1 1） wrMl2 and an element of order k ＋ 1.

The Automorphism Group of U（4,Z）

In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G_1×D_8×Z_2,where G_1=(a,b,c|a^4=b^2=c^2=1,a^b=a^(-1),[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z.

Rationality of quotients by linear actions of affine groups

LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable. We explicitly characterize two-step extensions 0 → S → V → Q → 0, with completely reducible S and Q, whose rationality cannot be obtained by the methods presented here (Theorem 5.3).

On Fundamental Group of a Certain Class of Welded Knots

In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.

A Construction Method of Cartesian Authentication Codes from B N Pair Decomposition of SL_n(F_q)

In the present paper, a construction of Cartesian authentication codes by using the BN pair decomposition of special linear group is presented. Moreover, under the case that the encoding rules are chosen according to a uniform probability distribution, the probability of a successful impersonation attack and the probability of a successful substitution attack of the code are computed.

Characteristic conditions-for strongly continuous groups of linear operators on a Hilbert space

IN 1932, Stone [1] gave the first result concerning C0 semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C0 group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations （PDEs） with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C0 group on a Banach space. There are also some conditions under which a C0 semigroup on a Banach space can be embedded in a C0 group. We refer the readers to ref. [3] for details of the results mentioned above.

Linear mathematical model for the unique solution of 3D ptychographic iterative engine

Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was developed to solve these linear equations for the transmission functions of all sample slices analytically.With both theoretical analysis and numerical simulations,this study revealed the underlying physics and mathematics of 3PIE and demonstrated for the first time,to our knowledge,that 3PIE can generate mathematically unique reconstruction even with noisy data.

Characterization of Groups L2(q) by NSE Where q ∈ {17, 27, 29}

The authors show that linear simple groups L_2(q) with q ∈ {17, 27, 29} can be uniquely determined by nse(L_2(q)), which is the set of numbers of elements with the same order.

Recognition of the Projective Special Linear Group over GF(3)

Let P be a finite group and denote by w（P） the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω（G） =ω（P） there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp（3）, where p 〉 3 is a prime number, is at most 2-recognizable.

Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.

t-Singular Linear Spaces

As a generalization of singular linear spaces, we introduce the concept of t-singular linear spaces, make some anzahl formulas of subspaces, and determine the suborbits of t-singular linear groups.