Centraliser near-rings determined by fixed point free automorphism groups

1987 ◽  
Vol 107 (3-4) ◽  
pp. 327-337 ◽  
Author(s):  
Peter Fuchs ◽  
C. J. Maxson ◽  
M. R. Pettet ◽  
K. C. Smith
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
Nontrivial Ideal

SynopsisLet G be a group and let A be a fixed point free group of automorphisms of G. It is shown that the centraliser near-ring MA(G) has at most one nontrivial ideal. Conditions on the pair (A, G) are given which force MA(G) to be simple. It is shown that if a nonsimple near-ring MA(G) exists, then A and G have unusual properties.

scholarly journals On soluble groups which admit the dihedral group of order eight fixed-point-freely

1973 ◽  
Vol 8 (2) ◽  
pp. 305-312 ◽  
Author(s):  
Alan R. Camina ◽  
F. Peter Lockett
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Soluble Group ◽  
Dihedral Group ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Nilpotent Length

If the finite soluble group G admits the dihedral group of order eight as a fixed-point-free group of automorphisms then the nilpotent length of G is at most three.

Half-Transitive Automorphism Groups

1966 ◽  
Vol 18 ◽  
pp. 1243-1250 ◽  
Author(s):  
I. M. Isaacs ◽  
D. S. Passman
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Permutation Group ◽  
Automorphism Groups ◽  
Permutation Representation ◽  
Group Of Automorphisms ◽  
Transitive Automorphism ◽  
A Finite Group ◽  
Special Case ◽  
Identity Elements

Let G be a finite group and A a group of automorphisms of G. Clearly A acts as a permutation group on G#, the set of non-identity elements of G. We assume that this permutation representation is half transitive, that is all the orbits have the same size. A special case of this occurs when A acts fixed point free on G. In this paper we study the remaining or non-fixed point free cases. We show first that G must be an elementary abelian g-group for some prime q and that A acts irreducibly on G. Then we classify all such occurrences in which A is a p-group.

scholarly journals The cup product of Brooks quasimorphisms

2018 ◽  
Vol 30 (5) ◽  
pp. 1157-1162 ◽  
Author(s):  
Michelle Bucher ◽  
Nicolas Monod
Keyword(s):  
Free Group ◽  
Automorphism Groups ◽  
Bounded Cohomology ◽  
Universal Class ◽  
Cup Product

AbstractWe prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.

Nilpotent Partition-Inducing Automorphism Groups

1981 ◽  
Vol 33 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
Identity Elements

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

Fixed Points of Automorphisms of Compact Riemann Surfaces

1970 ◽  
Vol 22 (5) ◽  
pp. 922-932 ◽  
Author(s):  
M. J. Moore
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Riemann Surfaces ◽  
Homology Group ◽  
Group Of Automorphisms ◽  
Quartic Curve ◽  
Lefschetz Fixed Point Formula ◽  
Compact Riemann Surfaces ◽  
Klein's Quartic Curve ◽  
Fundamental Paper

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

Automorphism groups and invariant theory on ℙN

2018 ◽  
Vol 17 (09) ◽  
pp. 1850162 ◽  
Author(s):  
João Alberto de Faria ◽  
Benjamin Hutz
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Invariant Theory ◽  
Automorphism Groups ◽  
Finite Subgroup ◽  
Fixed Point Algorithm ◽  
Projective Linear Group ◽  
Rationality Problems ◽  
Stabilizer Group ◽  
A Finite Group

Let [Formula: see text] be a field and [Formula: see text] a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group [Formula: see text]. The group of automorphisms, or stabilizer group, of a given [Formula: see text] for this action is known to be a finite group. In this paper, we apply methods of invariant theory to automorphism groups by addressing two mainly computational problems. First, given a finite subgroup of [Formula: see text], determine endomorphisms of [Formula: see text] with that group as a subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. Inversely, given an endomorphism, determine its automorphism group. In particular, we extend the Faber–Manes–Viray fixed-point algorithm for [Formula: see text] to endomorphisms of [Formula: see text]. A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.

Rings with Fixed-Point-Free Group Actions

1973 ◽  
Vol s3-27 (1) ◽  
pp. 69-87 ◽  
Author(s):  
G. M. Bergman ◽  
I. M. Isaacs
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Group Actions ◽  
Free Group Actions

scholarly journals On the influence of fixed point free nilpotent automorphism groups

2016 ◽  
Vol 184 (4) ◽  
pp. 531-538 ◽  
Author(s):  
Gülin Ercan ◽  
İsmail Ş. Güloğlu
Keyword(s):  
Fixed Point ◽  
Automorphism Groups
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