Locally finite minimal non-FC-groups

1989 ◽  
Vol 105 (3) ◽  
pp. 417-420 ◽  
Author(s):  
Mahmut Kuzucuoglu ◽  
Richard E. Phillips
Keyword(s):  
Simple Group ◽  
Proper Subgroup ◽  
Positive Answer ◽  
Locally Finite ◽  
Finite Set

We recall that a group G is an FC-group if for every x∈G the set of conjugates {xg|g∈G} is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC-group: such groups are called minimal non-FC-groups. Locally finite minimal non-FC-groups with (G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non-FC-group is either a simple group or a p-group for some prime p. Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the followingTheorem. There exists no simple, locally finite, minimal non-FC-group.

A characteristically simple group

1954 ◽  
Vol 50 (4) ◽  
pp. 641-642 ◽  
Author(s):  
D. H. McLain
Keyword(s):  
Simple Group ◽  
Finite Order ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Characteristic Subgroup ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Finite Set ◽  
Group A ◽  
Trivial Centre

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.

Nonrecursive tilings of the plane. I

1974 ◽  
Vol 39 (2) ◽  
pp. 283-285 ◽  
Author(s):  
William Hanf
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Positive Answer ◽  
Additional Restriction ◽  
Detailed History ◽  
Finite Set

A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.Given a finite set of tiles T1, …, Tn, we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

scholarly journals INTEGRAL GROUP RING OF THE MATHIEU SIMPLE GROUP M24

2012 ◽  
Vol 11 (01) ◽  
pp. 1250016 ◽  
Author(s):  
VICTOR BOVDI ◽  
ALEXANDER KONOVALOV
Keyword(s):  
Simple Group ◽  
Group Ring ◽  
Unit Group ◽  
Positive Answer ◽  
Integral Group Ring ◽  
Integral Group ◽  
Sporadic Group ◽  
Prime Graphs ◽  
Zassenhaus Conjecture

We study the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M24. As a consequence, for this group we give a positive answer to the question by Kimmerle about prime graphs.

A characterization of piecewise linear homology manifolds

1983 ◽  
Vol 93 (2) ◽  
pp. 271-274 ◽  
Author(s):  
W. J. R. Mitchell
Keyword(s):  
Piecewise Linear ◽  
Subsequent Paper ◽  
Locally Compact ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Set ◽  
Topological Manifolds ◽  
Homology Manifolds

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

scholarly journals Measures with locally finite support and spectrum

2016 ◽  
Vol 113 (12) ◽  
pp. 3152-3158 ◽  
Author(s):  
Yves F. Meyer
Keyword(s):  
Fourier Transform ◽  
Finite Support ◽  
Locally Finite ◽  
Aperiodic Order ◽  
Finite Set ◽  
Dirac Comb ◽  
The Fourier Transform ◽  
Insight Into

The goal of this paper is the construction of measures μ on Rn enjoying three conflicting but fortunately compatible properties: (i) μ is a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transform μ^ of μ is also a sum of weighted Dirac masses on a locally finite set, and (iii) μ is not a generalized Dirac comb. We give surprisingly simple examples of such measures. These unexpected patterns strongly differ from quasicrystals, they provide us with unusual Poisson's formulas, and they might give us an unconventional insight into aperiodic order.

Equations in Type-2 Fuzzy Sets

2015 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 31-42
Author(s):  
John Harding ◽  
Carol Walker ◽  
Elbert Walker
Keyword(s):  
Fuzzy Sets ◽  
Decision Procedure ◽  
Unit Interval ◽  
Main Concern ◽  
Locally Finite ◽  
Equational Basis ◽  
Finite Set ◽  
Positive Results ◽  
Truth Tables

The main concern of this paper is with the equations satisfied by the algebra of truth values of type-2 fuzzy sets. That algebra has elements all mappings from the unit interval into itself with operations given by certain convolutions of operations on the unit interval. There are a number of positive results. Among them is a decision procedure, similar to the method of truth tables, to determine when an equation holds in this algebra. One particular equation that holds in this algebra implies that every subalgebra of it that is a lattice is a distributive lattice. It is also shown that this algebra is locally finite. Many questions are left unanswered. For example, we do not know whether or not this algebra has a finite equational basis, that is, whether or not there is a finite set of equations from which all equations satisfied by this algebra follow. This and various other topics about the equations satisfied by this algebra will be discussed.

scholarly journals On the laws of the variety se

1972 ◽  
Vol 14 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Roger M. Bryant
Keyword(s):  
Positive Integer ◽  
Simple Group ◽  
Finite Simple Group ◽  
Basis Property ◽  
Finite Basis ◽  
Simple Groups ◽  
Locally Finite ◽  
Prove Theorem ◽  
Finite Basis Property ◽  
Locally Finite Variety

A group is called an s-group if it is locally finite and all its Sylow subgroups are abelian. Kovács [4] has shown that, for any positive integer e, the class se of all s-groups of exponent dividing e is a (locally finite) variety. The proof of this relies on the fact that, for any e, there are only finitely many (isomorphism classes of) non-abelian finite simple groups in se; and this is a consequence of deep results of Walter and others (see [6]). In [2], Christensen raised the finite basis question for the laws of the varieties se. It is easy to establish the finite basis property for an se which contains no non-abelian finite simple group; and Christensen gave a finite basis for the laws of the variety s30, whose only non-abelian finite simple group is PSL(2,5). Here we prove Theorem For any positive integer e, the varietysehas a finite basis for its laws.

scholarly journals A reduction theorem for perfect locally finite minimal non-FC groups

1999 ◽  
Vol 41 (1) ◽  
pp. 81-83 ◽  
Author(s):  
FELIX LEINEN
Keyword(s):  
Conjugacy Class ◽  
Proper Subgroup ◽  
Conjugacy Classes ◽  
The Other ◽  
Reduction Theorem ◽  
Locally Finite ◽  
Open Question ◽  
Finite Conjugacy

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

scholarly journals Reconstructing Subsets of Reals

10.37236/1452 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
A. J. Radcliffe ◽  
A. D. Scott
Keyword(s):  
Finite Subset ◽  
Fixed Size ◽  
Real Numbers ◽  
Locally Finite ◽  
Finite Set ◽  
Almost All

We consider the problem of reconstructing a set of real numbers up to translation from the multiset of its subsets of fixed size, given up to translation. This is impossible in general: for instance almost all subsets of $\mathbb{Z}$ contain infinitely many translates of every finite subset of $\mathbb{Z}$. We therefore restrict our attention to subsets of $\mathbb{R}$ which are locally finite; those which contain only finitely many translates of any given finite set of size at least 2. We prove that every locally finite subset of $\mathbb{R}$ is reconstructible from the multiset of its 3-subsets, given up to translation.

scholarly journals Centralizers of abelian subgroups in locally finite simple groups

1997 ◽  
Vol 40 (2) ◽  
pp. 217-225
Author(s):  
M. Kuzucuoǧlu
Keyword(s):  
Simple Group ◽  
Finite Length ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Locally Finite ◽  
Odd Order ◽  
Non Linear ◽  
Abelian Subgroups

It is shown that, if a non-linear locally finite simple group is a union of finite simple groups, then the centralizer of every element of odd order has a series of finite length with factors which are either locally solvable or non-abelian simple. Moreover, at least one of the factors is non-linear simple. This is also extended to abelian subgroup of odd orders.

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