scholarly journals The fixed point theorem for simplicial nonpositive curvature

2008 ◽  
Vol 144 (3) ◽  
pp. 683-695
Author(s):  
PIOTR PRZYTYCKI
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Nonpositive Curvature ◽  
Free Products ◽  
Locally Finite ◽  
Hnn Extensions ◽  
Finite Subgroups ◽  
The Fixed Point Theorem ◽  
A Finite Group ◽  
Free Products With Amalgamation

AbstractWe prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter ≤5. For 7–systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7–systolic groups over finite subgroups are also 7–systolic.

scholarly journals Locally finite approximation of Lie groups. II

1986 ◽  
Vol 100 (3) ◽  
pp. 505-517 ◽  
Author(s):  
Eric M. Friedlander ◽  
Guido Mislin
Keyword(s):  
Finite Group ◽  
Lie Group ◽  
Homotopy Class ◽  
Classifying Space ◽  
Locally Finite ◽  
Component Group ◽  
Finite Dimensional ◽  
Finite Subgroups ◽  
Finite Approximation ◽  
A Finite Group

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.

ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

2019 ◽  
Vol 102 (1) ◽  
pp. 77-90
Author(s):  
PABLO SPIGA
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Alternating Group ◽  
Permutation Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

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.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

scholarly journals Annular Dehn functions of groups

1998 ◽  
Vol 58 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Stephen G. Brick ◽  
Jon M. Corson
Keyword(s):  
Conjugacy Problem ◽  
Upper Bounds ◽  
Dehn Function ◽  
Free Products ◽  
Dehn Functions ◽  
Finitely Presented Group ◽  
Hnn Extensions ◽  
Finite Subgroups ◽  
Boundary Curves ◽  
Finitely Presented

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

scholarly journals Two bounds on the noncommuting graph

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Stefano Nardulli ◽  
Francesco G. Russo
Keyword(s):  
Finite Group ◽  
General Context ◽  
Sobolev Inequalities ◽  
Classical Analysis ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
A Finite Group ◽  
Noncommuting Graph ◽  
First Time

AbstractErdős introduced the noncommuting graph in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph has become relevant during the last years for various reasons. Here we deal with a numerical aspect, showing for the first time an isoperimetric inequality and an analytic condition in terms of Sobolev inequalities. This last result holds in the more general context of weighted locally finite graphs.

scholarly journals THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY

2021 ◽  
pp. 1-70
Author(s):  
DAVID GEPNER ◽  
JEREMIAH HELLER
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Homotopy Theory ◽  
Independent Interest ◽  
Splitting Theorem ◽  
Motivic Homotopy Theory ◽  
Group A ◽  
Motivic Homotopy ◽  
A Finite Group

Abstract We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

Splitting over nilpotent and hypercentral residuals

1975 ◽  
Vol 78 (2) ◽  
pp. 215-226 ◽  
Author(s):  
B. Hartley ◽  
M. J. Tomkinson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Formation Theory ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Nilpotent Residual ◽  
A Finite Group

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism

1968 ◽  
Vol 20 ◽  
pp. 1300-1307 ◽  
Author(s):  
Fletcher Gross
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Simple Group ◽  
Identity Element ◽  
Simple Groups ◽  
Cyclic Groups ◽  
A Finite Group ◽  
Open Question

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

scholarly journals Finite groups admitting fixed point free automorphisms of order pq

1987 ◽  
Vol 30 (1) ◽  
pp. 51-56 ◽  
Author(s):  
Cheng Kai-Nah
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Finite Groups ◽  
Fitting Height ◽  
A Finite Group

By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.

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