scholarly journals On a minimal counterexample to the Alperin-McKay conjecture

2011 ◽  
Vol 87 (10) ◽  
pp. 192-193 ◽  
Author(s):  
Masafumi Murai
Keyword(s):  
Minimal Counterexample

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

Generix never gives up

2006 ◽  
Vol 71 (2) ◽  
pp. 599-610 ◽  
Author(s):  
Eric Jaligot
Keyword(s):  
Morley Rank ◽  
Carter Subgroup ◽  
Minimal Counterexample ◽  
Finite Morley Rank

AbstractWe prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal counterexample to the Genericity Conjecture.

scholarly journals CONJUGACY OF CARTER SUBGROUPS IN GROUPS OF FINITE MORLEY RANK

2008 ◽  
Vol 08 (01) ◽  
pp. 41-92 ◽  
Author(s):  
OLIVIER FRÉCON
Keyword(s):  
Conjugacy Class ◽  
Algebraic Groups ◽  
Simple Groups ◽  
Morley Rank ◽  
Minimal Counterexample ◽  
Finite Morley Rank ◽  
Cartan Subgroups

The Cherlin–Zil'ber Conjecture states that all simple groups of finite Morley rank are algebraic. We prove that any minimal counterexample to this conjecture has a unique conjugacy class of Carter subgroups, which are analogous to Cartan subgroups in algebraic groups.

scholarly journals On a Minimal Counterexample to Dade's Projective Conjecture

2002 ◽  
Vol 249 (2) ◽  
pp. 453-462 ◽  
Author(s):  
Charles W. Eaton ◽  
Geoffrey R. Robinson
Keyword(s):  
Minimal Counterexample

Constructing a Minimal Counterexample in Group Theory

1985 ◽  
Vol 58 (1) ◽  
pp. 24
Author(s):  
Arnold D. Feldman
Keyword(s):  
Group Theory ◽  
Minimal Counterexample

The topological Vaught's conjecture and minimal counterexamples

1994 ◽  
Vol 59 (3) ◽  
pp. 757-784 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Topological Group ◽  
Polish Space ◽  
Sufficient Conditions ◽  
Usual Model ◽  
Borel Set ◽  
Isomorphism Classes ◽  
Perfect Set ◽  
Minimal Counterexample ◽  
Necessary And Sufficient ◽  
Borel Measurable

Let G be a Polish topological group, let X be a Polish space, let J: G × X → X be a Borel-measurable action of G on X, and let A ⊂ X be a Borel set which is invariant with respect to J, i.e., a Borel set of orbits. The following statement, or various equivalent versions of it, is known as the Topological Vaught's Conjecture.Let (G, X, J, A) be as above. Either A contains only countably many orbits, or else, A contains perfectly many orbits.We say that A contains perfectly many orbits if there is a perfect set P ⊂ A such that no two elements of P are in the same orbit. (Assuming ¬CH, A contains perfectly many orbits iff it contains 2ℵ0 orbits.) The Topological Vaught's Conjecture implies the usual, model theoretic, Vaught's Conjecture for Lω1ω, since the isomorphism classes are the orbits of an action of the group of permutations of ω; we give details in §0. The “Borel” assumption cannot be weakened for either A or J.Given a Borel-measurable Polish action (G,X,J) and an invariant Borel set B ⊂ X, we say that B is a minimal counterexample if (G,X,J,B) is a counterexample to the Topological Vaught's Conjecture and for every invariant Borel C ⊂ B, either C or B\C contains only countably many orbits. This paper is concerned with counterexamples to the Topological Vaught's Conjecture (of course, there may not be any), and in particular, with minimal counterexamples. First, there is a theorem on the existence of minimal counterexamples. This theorem was known for the model theoretic case (it is due to Harnik and Makkai), and is here generalized to arbitrary Borel-measurable Polish actions. Second, we study the properties of minimal counterexamples. We give two different necessary and sufficient conditions for a counterexample to be minimal, as well as some consequences of minimality. Some of these results are proved assuming determinacy axioms.This second part seems to be new even in the model theoretic case.

scholarly journals On minimal faithful permutation representations of finite groups

2000 ◽  
Vol 62 (2) ◽  
pp. 311-317 ◽  
Author(s):  
L. G. Kovács ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Symmetric Group ◽  
Finite Groups ◽  
Abelian Normal Subgroup ◽  
Minimal Counterexample ◽  
Representations Of Finite Groups ◽  
A Finite Group

The minimal faithful permutation degree μ(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let N be a normal subgroup of a finite group G. We prove that μ(G/N) ≤ μ(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.

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