On the order of the automorphism group of a finite group. II

1956 ◽  
Vol 235 (1201) ◽  
pp. 235-246 ◽  
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Group Ii ◽  
A Finite Group

Corresponding to a fixed prime p there exists a function g(h) such that the order of the automorphism group of a finite group G is divisible by p h , provided that the order of G is divisible by p g(h) . An explicit upper bound for g(h) is obtained.

On the Minimum Order of Graphs with Given Group

1974 ◽  
Vol 17 (4) ◽  
pp. 467-470 ◽  
Author(s):  
László Babai
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Minimum Order ◽  
Number Of Generators ◽  
Minimum Number ◽  
A Finite Group

For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).

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.

On the number of conjugacy classes in a finite group II

1988 ◽  
Vol 64 (1) ◽  
pp. 87-127 ◽  
Author(s):  
Antonio Vera-López ◽  
MA Concepción Larrea
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Group Ii ◽  
A 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 Abelian Sylow subgroups in a finite group, II

2015 ◽  
Vol 421 ◽  
pp. 3-11 ◽  
Author(s):  
Gabriel Navarro ◽  
Ronald Solomon ◽  
Pham Huu Tiep
Keyword(s):  
Finite Group ◽  
Sylow Subgroups ◽  
Group Ii ◽  
A Finite Group

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.

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.

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