scholarly journals Every finite group is the automorphism group of some perfect code

1986 ◽  
Vol 43 (1) ◽  
pp. 45-51 ◽  
Author(s):  
K.T Phelps
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Perfect Code

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

Tournaments With a Given Automorphism Group

1964 ◽  
Vol 16 ◽  
pp. 485-489 ◽  
Author(s):  
J. W. Moon
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Oriented Graph ◽  
Not Given ◽  
Oriented Graphs ◽  
Vertex Set ◽  
A Finite Group ◽  
Graph Form

The set of all adjacency-preserving automorphisms of the vertex set of a graph form a group which is called the (automorphism) group of the graph. In 1938 Frucht (2) showed that every finite group is isomorphic to the group of some graph. Since then Frucht, Izbicki, and Sabidussi have considered various other properties that a graph having a given group may possess. (For pertinent references and definitions not given here see Ore (4).) The object in this paper is to treat by similar methods a corresponding problem for a class of oriented graphs. It will be shown that a finite group is isomorphic to the group of some complete oriented graph if and only if it has an odd number of elements.

scholarly journals Sabidussi-type theorems for stability

1974 ◽  
Vol 10 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Douglas D. Grant ◽  
D.A. Holton ◽  
K.L. McAvaney
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Abstract Group ◽  
The Given

In this note we give details of a method by which we can produce an index-0 graph from any unstable graph and use it to show that given any finite group there exists an index-0 graph whose automorphism group is isomorphic, as an abstract group, to the given group. We proceed to construct two infinite families of connected index-0 graphs with connected complements whose automorphism group contains a transposition. This enables us to produce, for any finite group G, an index-0 graph whose automorphism group, isomorphic as an abstract group to C2 × G, contains a transposition.

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].

scholarly journals On groups admitting a noncyclic abelian automorphism group

1973 ◽  
Vol 9 (3) ◽  
pp. 363-366 ◽  
Author(s):  
J.N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Automorphism Group ◽  
Fixed Points ◽  
Free Operator ◽  
A Finite Group

It is shown that a condition of Kurzwell concerning fixed-points of certain operators on a finite group G is sufficient to ensure that G is soluble. The result generalizes those of Martineau on elementary abelian fixed-point-free operator groups.

scholarly journals Cubic identity graphs and planar graphs derived from trees

1970 ◽  
Vol 11 (2) ◽  
pp. 207-215 ◽  
Author(s):  
A. T. Balaban ◽  
Roy O. Davies ◽  
Frank Harary ◽  
Anthony Hill ◽  
Roy Westwick
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Planar Graphs ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Plane Trees ◽  
Nontrivial Identity

AbstractThe smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

Groups of real genus p + 1, where p is prime

2014 ◽  
Vol 14 (03) ◽  
pp. 1550040
Author(s):  
Coy L. May
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Klein Surface ◽  
The Real ◽  
A Finite Group ◽  
Large Groups

Let G be a finite group. The real genusρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We classify the large groups of real genus p + 1, that is, the groups such that |G| ≥ 3(g - 1), where the genus action of G is on a bordered surface of genus g = p + 1. The group G must belong to one of four infinite families. In addition, we determine the order of the largest automorphism group of a surface of genus g for all g such that g = p + 1, where p is a prime.

scholarly journals On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

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