Characters of nilpotent groups

1984 ◽  
Vol 96 (1) ◽  
pp. 123-137 ◽  
Author(s):  
A. L Carey ◽  
W. Moran
Keyword(s):  
Conjugacy Class ◽  
Large Class ◽  
Nilpotent Groups ◽  
Positive Definite ◽  
Primitive Ideals ◽  
Finite Conjugacy ◽  
Finite Conjugacy Class ◽  
The Relationship

AbstractThe characters (extremal positive definite central functions) of discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group C*-algebra C*(G) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1–1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.

scholarly journals Property RD and the classification of traces on reduced group C∗-algebras of hyperbolic groups

2017 ◽  
Vol 09 (04) ◽  
pp. 707-716
Author(s):  
Sherry Gong
Keyword(s):  
Conjugacy Class ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Linear Combinations ◽  
Rapid Decay ◽  
C Algebras ◽  
Rapid Decay Property ◽  
Finite Conjugacy ◽  
Finite Conjugacy Class

In this paper, we show that if [Formula: see text] is a non-elementary word hyperbolic group, [Formula: see text] is an element, and the conjugacy class of [Formula: see text] is infinite, then all traces [Formula: see text] vanish on [Formula: see text]. Moreover, we completely classify all traces by showing that traces [Formula: see text] are linear combinations of traces [Formula: see text] given by [Formula: see text] where [Formula: see text] is an element with finite conjugacy class, denoted [Formula: see text]. We demonstrate these two statements by introducing a new method to study traces that uses Sobolev norms and the rapid decay property.

Localization, Algebraic Loops and H-Spaces I

1979 ◽  
Vol 31 (2) ◽  
pp. 427-435 ◽  
Author(s):  
Albert O. Shar
Keyword(s):  
Nilpotent Group ◽  
Algebraic Structure ◽  
Nilpotent Groups ◽  
Topological Spaces ◽  
Finitely Generated ◽  
Cw Complex ◽  
Topological Localization ◽  
The Relationship

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.

scholarly journals Subnomality in soluble minimax groups

1974 ◽  
Vol 17 (1) ◽  
pp. 113-128 ◽  
Author(s):  
D. J. McCaughan
Keyword(s):  
Large Class ◽  
Upper Bound ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Wreath Products ◽  
Minimal Length ◽  
Intersection Property ◽  
Finite Chain ◽  
Subnormal Subgroups

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

On various types of convergence of positive definite functions on foundation semigroups

1992 ◽  
Vol 111 (2) ◽  
pp. 325-330 ◽  
Author(s):  
M. Lashkarizadeh-Bami
Keyword(s):  
Compact Group ◽  
Pointwise Convergence ◽  
Positive Definite ◽  
Limit Problem ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Interesting Discussion ◽  
Central Limit Problem ◽  
The Relationship

As is known, on a locally compact group G, the mere assumption of pointwise convergence of a sequence (n) of continuous positive definite functions implies uniform convergence of (n) to on compact subsets of G. This result was first proved in 1947 by Raikov8 (and independently by Yoshizawa9). An interesting discussion of the relationship between such theorems and various Cramr-Lvy theorems of the 1920s and 1930s, concerning the Central Limit Problem of probability, is given by McKennon(7, p. 62).

scholarly journals FINITE CONJUGACY IN ALGEBRAS AND ORDERS

2001 ◽  
Vol 44 (1) ◽  
pp. 201-213 ◽  
Author(s):  
M. A. Dokuchaev ◽  
S. O. Juriaans ◽  
C. Polcino Milies ◽  
M. L. Sobral Singer
Keyword(s):  
Conjugacy Class ◽  
Division Ring ◽  
Multiplicative Group ◽  
Central Element ◽  
Subject Classification ◽  
Group Rings ◽  
2000 Mathematics Subject Classification ◽  
Primary 16U60 ◽  
Finite Conjugacy

AbstractHerstein showed that the conjugacy class of a non-central element in the multiplicative group of a division ring is infinite. We prove similar results for units in algebras and orders and give applications to group rings.AMS 2000 Mathematics subject classification: Primary 16U60. Secondary 16H05; 16S34; 20F24; 20C05

Weak Cayley tables and generalized centralizer rings of finite groups

2012 ◽  
Vol 153 (2) ◽  
pp. 281-318 ◽  
Author(s):  
STEPHEN P. HUMPHRIES ◽  
EMMA L. RODE
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Cayley Table ◽  
A Finite Group ◽  
Finite Conjugacy ◽  
Relationship Of ◽  
Cayley Tables ◽  
The Relationship

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

scholarly journals Nilpotent injectors in finite groups

1985 ◽  
Vol 32 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Peter Förster
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Fitting Class ◽  
Basic Properties ◽  
A Finite Group ◽  
Class F

Nilpotent injectors exist in all finite groups.For every Fitting class F of finite groups (see [2]), InjF(G) denotes the set of all H ≤ G such that for each N ⊴ ⊴ G , H ∩ N is an F -maximal subgroup of N (that is, belongs to F and i s maximal among the subgroups of N with this property). Let W and N* denote the Fitting class of all nilpotent and quasi-nilpotent groups, respectively. (For the basic properties of quasi-nilpotent groups, and of the N*-radical F*(G) of a finite group G3 the reader is referred to [5].,X. %13; we shall use these properties without further reference.) Blessenohl and H. Laue have shown in CJ] that for every finite group G, InjN*(G) = {H ≤ G | H ≥ F*(G) N*-maximal in G} is a non-empty conjugacy class of subgroups of G. More recently, Iranzo and Perez-Monasor have verified InjN(G) ≠ Φ for all finite groups G satisfying G = CG(E(G))E(G) (see [6]), and have extended this result to a somewhat larger class M of finite groups C(see [7]). One checks, however, that M does not contain all finite groups; for example, S5 ε M.

scholarly journals Conformality and p-isomorphism in finite nilpotent groups

1967 ◽  
Vol 7 (2) ◽  
pp. 165-171 ◽  
Author(s):  
C. D. H. Cooper
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Equivalence Relations ◽  
The Relationship

This paper discusses the relationship between two equivalence relations on the class of finite nilpotent groups. Two finite groups are conformal if they have the same number of elements of all orders. (Notation: G ≈ H.) This relation is discussed in [4] pp 107–109 where it is shown that conformality does not necessarily imply isomorphism, even if one of the groups is abelian. However, if both groups are abelian the position is much simpler. Finite conformal abelian groups are isomorphic.

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.

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