Invariants of Finite Groups over Finite Fields: Recent Progress and New Conjectures

2002 ◽  
pp. 112-122
Author(s):  
Peter Fleischmann
Keyword(s):  
Finite Fields ◽  
Finite Groups ◽  
Recent Progress

Factorizations of Finite Groups and Related Topics

2020 ◽  
Vol 27 (01) ◽  
pp. 149-180
Author(s):  
Lev Kazarin
Keyword(s):  
Finite Groups ◽  
Recent Progress ◽  
Wide Area ◽  
Nilpotent Algebras ◽  
Fundamental Paper

This is a survey on the recent progress in the theory of finite groups with factorizations and around it, done by the author and his co-authors, and this has no pretensions to cover all topics in this wide area of research. In particular, we only touch the great consequences of the fundamental paper of Liebeck, Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations. In each case the reader can find additional references at the end of Section 1. Some of the methods of investigation can be used to obtain information about finite groups in general, nilpotent algebras and related nearrings.

T2-Groups And a Characterization of the Finite Groups of Moebius Transformations

1965 ◽  
Vol 17 ◽  
pp. 353-366 ◽  
Author(s):  
P. J. Lorimer
Keyword(s):  
Finite Fields ◽  
Finite Groups ◽  
The Real ◽  
Real Complex

In recent years a number of algebraic characterizations of the groups of Moebius transformations over finite fields have been given in the literature; see (1, 3, 6). H. W. E. Schwerdtfeger has noticed (4) that the group G of Moebius transformations over the real, complex, and certain other fields has the property:

On Finite Groups and Finite Fields

1991 ◽  
Vol 98 (6) ◽  
pp. 549
Author(s):  
J. D. Reid
Keyword(s):  
Finite Fields ◽  
Finite Groups

On Finite Groups and Finite Fields

1991 ◽  
Vol 98 (6) ◽  
pp. 549-551
Author(s):  
J. D. Reid
Keyword(s):  
Finite Fields ◽  
Finite Groups

scholarly journals Pseudofinite structures and simplicity

2015 ◽  
Vol 15 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Darío García ◽  
Dugald Macpherson ◽  
Charles Steinhorn
Keyword(s):  
Finite Fields ◽  
Finite Groups ◽  
Local Stability ◽  
Vector Spaces ◽  
Regularity Lemma ◽  
Suitable Assumption ◽  
Underlying Theory ◽  
Finite Structures

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

scholarly journals Existence of finite groups with classical commutator subgroup

1978 ◽  
Vol 25 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Michael D. Miller
Keyword(s):  
Abelian Group ◽  
Finite Fields ◽  
Finite Groups ◽  
Commutator Subgroup ◽  
Unitary Groups ◽  
The Other ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Orthogonal Groups ◽  
General Linear Groups

AbstractGiven a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

scholarly journals DEFINABLY SIMPLE STABLE GROUPS WITH FINITARY GROUPS OF AUTOMORPHISMS

2019 ◽  
Vol 84 (02) ◽  
pp. 704-712
Author(s):  
ULLA KARHUMÄKI
Keyword(s):  
Finite Fields ◽  
Finite Groups ◽  
Simple Groups ◽  
Stable Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Groups Of Automorphisms ◽  
Groups Of Lie Type

AbstractWe prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps, and allow us to classify definably simple stable groups in the specific case when they admit such automorphisms.

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