MULTIPLICATIVE GROUP AUTOMORPHISMS OF INVERTIBLE UPPER TRIANGULAR MATRICES OVER FIELDS

Suppose F is a field of characteristic not 2 and F* its multiplicative group. Let T*n(F) be the multiplicative group of invertible upper triangular n x n matrices over F and STn(F) its subgroup {(aij) E T*n(F)aii = 1, i}. This paper proves that f: T*n(F) → T*n(F) is a group automorphism if and only if there exist a matrix Q in T*n(F) and a field automorphism rs of F such that either where A = ((aij)), A-T is the transpose inverse of A, J = Ei n+1-i, and : i= 1T*n(F) → F* is a homomorphism which satisfies {(xIn)(x)x F*} = F* and {x F*(xIn)(x) = 1} = {1}. Simultaneously, they also determine the automorphisms of STn(F).

The Orders of Automorphism Group of p^6 Rank Groups

In this paper, we determine the order of automorphism group of p-groups in the third family ( Φ 3) and the fourth family ( Φ 4) in [1], whose order is p^6(p≥3). Here p denotes an odd prime.

Finite Abelian Group of | A(G) |= 27pq

we have discussed structures of Abelian group G by order |A(G) |of automoorphism group and have obtained all types of finite Abelian grooup G when the order of A(G) equals 27pq(p,q are odd primmes).

The Automorphism Group of the Lie Group T（D（V_N,F））

In this paper,the authors determine maximal connected automorphism group of the Lie transformation group T（D（VN,F））,which acting on the normal Siegel domain D（VN,F）is simple and transitive,and prove that the maximal connected automorphism group of T（D（VN,F））is its maximal connected inner automorphism group.

THE HOLOMORPHIC AUTOMORPHISM GROUP OF HIGHER DIMENSION THULLEN DOMAIN

In this paper, we give the holomorphic automorphism group of the higher-dimensional generalization of Thullen domain.

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-

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.

Orders of Automorphism Groups of Some p-Groups

The orders of automorphism groups of the groups of order p^6 in the twelve family Ф12 axe produced, where p is an odd prime. Every group is analysed by utilizing the properties of metabelian, regularity and p-commutativity of finite p-groups, and the structure of the generators of its automorphism groups is obtained. Then the orders of automorphism groups are determined through some properties of equivalence in number theory.

The Bergman Kernel on Cartan-egg Domains of the First Type

We obtain the Bergman kernel and holomorphic automorphism group in explicit formulas on the Cartan egg domains of the first type.

The Bergman Kernels on WIII

In this paper, we compute the Bergman kernel function on WIII.and RIII(q) denote the Cartan domain of the third class. Because domain WIII is neither homogeneous domain nor Reinhardt domain, we will use a new way to solve this problem. First, we give a holomorphic automorphism group, such that for any Zo, there exists an element of this group, which maps (W, Zo) into (W,O). Second, introduce the concept of semi-Reinhardt and discuss the complete orthonormal system of this domain.

Isomorphism Between Gal(E/F) and Gal(L(E)/L(F))

Let E be a field of finite extension of a perfect field F. We show that Gal(E/F) is isomorphic to Gal(L(E)/L(F)).

Squeezing in Representation by Quadratic Combinations of Canonical Operators

The technique of integration within an ordered product of operators and the coherent-state representation are used to convert exponential operators of basis operators (P2, Q2, PQ + QP) to those of the basis operators (a2, a?2, a?a). The coherent state representation of unitary squeezing operators in the factorized form and their normal product form are thus derived. The squeezing engendered by operators of the general form is also obtained.

Finite dimensional factor algebras of F_(2)[X_(1),…,X_(n)]and their fixed point subalgebras

Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the finite field F2 (with respect to all F2-algebra automorphisms) are fully described.

The Automorphism Groups of Complete Lie Algebras with Commutative Nilpotent Radical

The automorphism groups of finite-dimensional complete Lie algebras withcommutative nilpotent radical over complex field C are given.

AUTOMORPHISM GROUPS OF 4-VALENT CONNECTED CAYLEY GRAPHS OF p-GROUPS

Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product GR x Ant(G,S), where GR is the right regular representation of G and Aut(G,S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a counterexample.

Automorphism group of Green ring of Sweedler Hopf algebra

Let H2 be Sweedler＇s 4-dimensional Hopf algebra and r（H2） be the corresponding Green ring of H2. In this paper, we investigate the automorphism groups of Green ring r（H2） and Green algebra F（H2） = r（H2） zF, where F is a field, whose characteristics is not equal to 2. We prove that the automorphism group of r（H2） is isomorphic to K4, where K4 is the Klein group, and the automorphism group of F（H2） is the semidirect product of Z2 and G, where G = F ／ {1/2} with multiplication given by a. b = 1 - a - b ＋ 2ab.

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p,where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p, where p is a prime number. As a consequence we prove if IGI = 25p, δ = 0, 1, 2 and p prime, then F 1 Cay（G, S） is a connected normal 1/2 arc-transitive Cayley graph only if G = F4p, where S is an inverse closed generating subset of G which does not contain the identity element of G and F4p is a group with presentation F4p = （a, b ｜aP = b4 = 1, b-lab = a^λ）, where λ2 = -1 （mod p）.

Graded automorphism group of TKK algebra

The classification of extended affine Lie algebras of type A 1 depends on the Tits-Kantor-Koecher (TKK) algebras constructed from semilattices of Euclidean spaces. One can define a unitary Jordan algebra J(S) from a semilattice S of ?v (v ≥ 1), and then construct an extended affine Lie algebra of type A 1 from the TKK algebra T(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction. In ?2 there are only two non-similar semilattices S and S’, where S is a lattice and S’ is a non-lattice semilattice. In this paper we study the ?2-graded automorphisms of the TKK algebra T(J(S)).

Derivation algebra and automorphism group of generalized topological N = 2 superconformal algebra

We determine the derivation algebra and the automorphism group of the generalized topological N = 2 superconformal algebra.

FINITE GROUPS WHOSE AUTOMORPHISM GROUP HAS ORDER CUBEFREE

LetGdenote a finite group.It is shown that if|Aut(G)|is cubefree thenGpossesses the Sylow tower property.