scholarly journals Automorphisms and Inner Automorphisms

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Ameer Jaber ◽  
Moh’D Yasein
Keyword(s):  
Classical Theorem ◽  
Group Of Automorphisms ◽  
Simple Superalgebra ◽  
Groups Of Automorphisms ◽  
Central Simple ◽  
Inner Automorphisms

LetKbe a field of characteristic not2and letA=A0+A1be central simple superalgebra overK, and let⁎be superinvolution onA. Our main purpose is to classify the group of automorphisms and inner automorphisms of(A,⁎)(i.e., commuting with⁎) by using the classical theorem of Skolem-Noether. Also we study two examples of groups of automorphisms and inner automorphisms on even central simple superalgebras with superinvolutions.

scholarly journals THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003

2013 ◽  
Vol 23 (06) ◽  
pp. 1485-1496 ◽  
Author(s):  
V. S. ATABEKYAN
Keyword(s):  
Normal Subgroup ◽  
Free Groups ◽  
Group Of Automorphisms ◽  
Burnside Groups ◽  
Free Burnside Group ◽  
Trivial Center ◽  
Groups Of Automorphisms ◽  
Group B ◽  
Inner Automorphisms ◽  
Automorphism Tower

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

Groups and Monoids of Regular Graphs (And of Graphs with Bounded Degrees)

1973 ◽  
Vol 25 (2) ◽  
pp. 239-251 ◽  
Author(s):  
Pavol Hell ◽  
Jaroslav Nešetřil
Keyword(s):  
Category Theory ◽  
Main Question ◽  
Regular Graphs ◽  
Group Of Automorphisms ◽  
Automorphisms Of Graphs ◽  
Usual Manner ◽  
The Subject ◽  
Groups Of Automorphisms

A graph X is a set V(X) (the vertices of X) with a system E(X) of 2-element subsets of V(X) (the edges of X). Let X, Y be graphs and f : V(X) → V(Y) a mapping; then/ is called a homomorphism of X into F if [f(x),f(y)] ∈ E(Y) whenever [x,y] ∈ E(X). Endomorphisms, isomorphisms and automorphisms are defined in the usual manner.Much work has been done on the subject of representing groups as groups of automorphisms of graphs (i.e., given a group G, to find a graph X such that the group of automorphisms of X is isomorphic to G). Recently, this was related to category theory, the main question being as to whether every monoid (i.e., semigroup with 1) can be represented as the monoid of endomorphisms of some graph in a given category of graphs.

scholarly journals Automorphisms of compact non-orientable Riemann surfaces

1971 ◽  
Vol 12 (1) ◽  
pp. 50-59 ◽  
Author(s):  
D. Singerman
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Riemann Surfaces ◽  
Group Of Automorphisms ◽  
Groups Of Automorphisms ◽  
Definition Of ◽  
Orientable Surfaces

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.

Groups of automorphisms and centralizers

1990 ◽  
Vol 107 (2) ◽  
pp. 227-238 ◽  
Author(s):  
Alexandre Turull
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Group Of Automorphisms ◽  
Fitting Height ◽  
Groups Of Automorphisms ◽  
The Difference

Let G be a finite solvable group and A a group of automorphisms of G such that (|A|, |G|) = 1. We denote by h(G) the Fitting height of G and by l(A) the length of the longest chain of subgroups of A. Then, under some additional hypotheses, it is known from [5] that h(G) ≤ 2l(A) + h(CG(A)) and from [8] that, when CG(A) = 1, h(G) ≤ l(A), both results being best possible (see [6, 7]). The present paper attempts to explain the difference in the coefficient of l(A) in the two inequalities, from 2 to 1.

ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1085-1106 ◽  
Author(s):  
G. MASHEVITZKY ◽  
B. I. PLOTKIN
Keyword(s):  
Lie Algebras ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Endomorphism Semigroup ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Inner Automorphism ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

Formation theory and groups of automorphisms of -groups

1977 ◽  
Vol 76 (4) ◽  
pp. 255-265 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Composition Series ◽  
Formation Theory ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Group Of Automorphisms ◽  
Infinite Case ◽  
Groups Of Automorphisms ◽  
Automorphisms Of Groups

SynopsisFurther results from the theory of finite soluble groups are extended to the class of locally finite groups with a satisfactory Sylow structure. Let be a saturated U-formation and A a -group of automorphisms of the -group G. A is said to act -centrally on G if G has an A-composition series (Λσ/Vσ; σ ∈ ∑) such that A induces an f(p)-group of automorphisms in each p-factor Λσ/Vσ. We show that in this situation A is an -group, thus generalising the result of Schmid [8]. Associated results of Schmid and of Baer are also extended to the infinite case.

scholarly journals NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP

2009 ◽  
Vol 52 (1) ◽  
pp. 169-177
Author(s):  
GÉRARD ENDIMIONI
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms ◽  
Free Soluble Group

AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

scholarly journals Outer automorphisms of hypercentral p-groups

1995 ◽  
Vol 37 (2) ◽  
pp. 243-247
Author(s):  
Orazio Puglisi
Keyword(s):  
Nilpotent Group ◽  
Full Group ◽  
Finite Nilpotent Group ◽  
Group Of Automorphisms ◽  
Outer Automorphisms ◽  
Inner Automorphisms

In his celebrated paper [3] Gaschiitz proved that any finite non-cyclic p-group always admits non-inner automorphisms of order a power of p. In particular this implies that, if G is a finite nilpotent group of order bigger than 2, then Out (G) = Aut(G)/Inn(G) =≠1. Here, as usual, we denote by Aut (G) the full group of automorphisms of G while Inn (G) stands for the group of inner automorphisms, that is automorphisms induced by conjugation by elements of G. After Gaschiitz proved this result, the following question was raised: “if G is an infinite nilpotent group, is it always true that Out (G)≠1?”

scholarly journals Derivations in central separable algebras

1978 ◽  
Vol 19 (1) ◽  
pp. 75-77 ◽  
Author(s):  
George T. Georgantas
Keyword(s):  
Galois Group ◽  
Brauer Group ◽  
Normal Extension ◽  
Noether Theorem ◽  
Central Simple Algebras ◽  
Group Extensions ◽  
Simple Algebras ◽  
Central Simple ◽  
Inner Automorphisms ◽  
Separable Algebras

Given N a finite separable normal extension of a field F, it is well known that the Brauer group Br(N/F) of classes of central simple F-algebras split by N is isomorphic with Ext(N*, G), the classes of group extensions of N* by the Galois group G of N over F. In the construction of this isomorphism, a key role is played by the Skolem-Noether Theorem which extends automorphisms to inner automorphisms in central simple algebras.

scholarly journals Local endomorphism near-rings

1988 ◽  
Vol 31 (3) ◽  
pp. 409-414 ◽  
Author(s):  
Carter G. Lyons ◽  
Gary L. Peterson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Group Of Automorphisms ◽  
The Third ◽  
Inner Automorphisms ◽  
A Finite Group ◽  
Generalized Quaternion

The purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups.

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