Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

PROBABILISTIC GENERATION OF WREATH PRODUCTS OF NON-ABELIAN FINITE SIMPLE GROUPS, II

2006 ◽  
Vol 16 (03) ◽  
pp. 493-503 ◽  
Author(s):  
MARTYN QUICK
Keyword(s):  
Simple Group ◽  
Wreath Product ◽  
Inverse Limit ◽  
Wreath Products ◽  
Profinite Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finitely Generated

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

Direct Cancellation of Finite Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 563-566
Author(s):  
Andrew Fransman ◽  
Peter Witbooi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
A Finite Group

We prove that if F is a finite group, and G and H are finitely generated groups with finite commutator subgroups for which F × G ≃ F × H, then G ≃ H.

scholarly journals ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 273-284 ◽  
Author(s):  
David J. Benson
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Cohomology Ring ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Characteristic P ◽  
Residue Classes ◽  
The Difference ◽  
A Finite Group

AbstractLet $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

EMBEDDABILITY OF GENERALISED WREATH PRODUCTS

2014 ◽  
Vol 91 (2) ◽  
pp. 250-263 ◽  
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Hilbert Space ◽  
Wreath Product ◽  
Metric Spaces ◽  
Wreath Products ◽  
Finitely Generated ◽  
Product Construction ◽  
Finitely Generated Groups

AbstractGiven two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

scholarly journals Bounds on the Action Degree of Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 271-282
Author(s):  
S. Alrehaili ◽  
Charef Beddani
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Random Element ◽  
General Relation ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
A Finite Group

The commutativity degree is the probability that a pair of elements chosen randomly from a group commute. The concept of  commutativity degree has been widely discussed by several authors in many directions.  One of the important generalizations of commutativity degree is the probability that a random element from a finite group G fixes a random element from a non-empty set S that we call the action degree of groups. In this research, the concept of action degree is further studied where some inequalities and bounds on the action degree of finite groups are determined.  Moreover, a general relation between the action degree of a finite group G and a subgroup H is provided. Next, the action degree for the direct product of two finite groups is determined. Previously, the action degree was only defined for finite groups, the action degree for finitely generated groups will be defined in this research and some bounds on them are going to be determined.

A NOTE ON THE NORMS OF TRANSITION OPERATORS ON LAMPLIGHTER GRAPHS AND GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1261-1272 ◽  
Author(s):  
WOLFGANG WOESS
Keyword(s):  
Random Walk ◽  
Wreath Products ◽  
Locally Compact Group ◽  
Operator Matrix ◽  
Locally Compact ◽  
Finitely Generated Groups ◽  
Group Of Isometries ◽  
Transition Operators ◽  
Special Case ◽  
Product Of Groups

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓp-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ2-spectral radius of symmetric random walks on such groups.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

On the Morava K-theory of some finite 2-groups

1997 ◽  
Vol 121 (1) ◽  
pp. 7-13 ◽  
Author(s):  
BJÖRN SCHUSTER
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Wreath Products ◽  
Classifying Space ◽  
Generalized Cohomology ◽  
K Theory ◽  
A Finite Group ◽  
Generalized Quaternion

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

scholarly journals Wreath Product Action on Generalized Boolean Algebras

10.37236/4831 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ashish Mishra ◽  
Murali K. Srinivasan
Keyword(s):  
Finite Group ◽  
Boolean Algebra ◽  
Wreath Product ◽  
Boolean Algebras ◽  
Product Action ◽  
Finite Set ◽  
A Finite Group ◽  
Multiplicity Free ◽  
Generalized Boolean Algebra ◽  
Generalized Boolean Algebras

Let $G$ be a finite group acting on the finite set $X$ such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product $G\sim S_n$ on the generalized Boolean algebra $B_X(n)$. We explicitly block diagonalize the commutant of this action.

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