scholarly journals The Number of Hall π-Subgroups of a π-Separable Group

2016 ◽  
Vol 06 (03) ◽  
pp. 162-165
Author(s):  
树珍 方
Keyword(s):  
Separable Group

The Character Correspondences on π-Separable Groups

2012 ◽  
Vol 19 (03) ◽  
pp. 501-508
Author(s):  
Huabin Si ◽  
Jiwen Zeng
Keyword(s):  
Separable Group ◽  
Irreducible Characters ◽  
The Relationship

In this paper, we mainly consider the relationship between the complex irreducible characters of a π-separable group and the complex irreducible characters of its Hall π-subgroup. If a π-group S acts on a π-separable group G, let H be an S-invariant Hall π-subgroup of G and CNG(H)/H(S)=1. Then we construct a natural bijection from the set Lin S(H) onto the set Irr π′,S(G). Furthermore, we get a bijection from the linear characters of H onto Irr π′(G).

Partial characters with respect to a normal subgroup

1999 ◽  
Vol 66 (1) ◽  
pp. 104-124 ◽  
Author(s):  
Gabriel Navarro ◽  
Lucia Sanus
Keyword(s):  
Normal Subgroup ◽  
Complex Space ◽  
Canonical Basis ◽  
Separable Group

AbstractSuppose that G is a π-separable group. Let N be a normal π1-subgroup of G and let H be a Hall π-subgroup of G. In this paper, we prove that there is a canonical basis of the complex space of the class functions of G which vanish of G-conjugates ofHN. When N = 1 and π is the complement of a prime p, these bases are the projective indecomposable characters and set of irreduciblt Brauer charcters of G.

scholarly journals On separable extensions of group rings and quaternion rings

1978 ◽  
Vol 1 (4) ◽  
pp. 433-438
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Full Description ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Ring Extension ◽  
Quaternion Algebras ◽  
Separable Group ◽  
Necessary And Sufficient ◽  
Separable Extensions

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extensionRG(Rmay be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extensionRQover a ringR, whereQare the usual quaternionsi,j,kand multiplication and addition are defined as quaternion algebras over a field. We shall show thatRGhas a unique separable idempotent if and only ifGis abelian, that there are more than one separable idempotents for a separable quaternion ringRQ, and thatRQis separable if and only if2is invertible inR.

scholarly journals On π-Block Induction in a π-Separable Group

2001 ◽  
Vol 235 (1) ◽  
pp. 261-266 ◽  
Author(s):  
Zhu Yixin
Keyword(s):  
Separable Group

OBTAINING NUCLEI FROM CHAINS OF NORMAL SUBGROUPS

2006 ◽  
Vol 05 (02) ◽  
pp. 215-229 ◽  
Author(s):  
MARK L. LEWIS
Keyword(s):  
Normal Subgroup ◽  
Irreducible Character ◽  
Normal Subgroups ◽  
Separable Group ◽  
Key Concepts ◽  
A Chain

In this paper, we reexamine the foundation of Isaacs' π-theory. One of the key concepts in Isaacs' π-theory is the construction of the characters Bπ(G) for a π-separable group G. The key to determining which characters lie in Bπ(G) was the construction of a nucleus for each irreducible character χ. In this paper, we present a different way of finding a nucleus for χ which is based on a chain of normal subgroup [Formula: see text]. Using this nucleus, we obtain the set of characters [Formula: see text]. We investigate the properties that [Formula: see text] has in common with Bπ(G).

scholarly journals CONSEQUENCES OF THE EXISTENCE OF AMPLE GENERICS AND AUTOMORPHISM GROUPS OF HOMOGENEOUS METRIC STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 876-886 ◽  
Author(s):  
MACIEJ MALICKI
Keyword(s):  
Hilbert Space ◽  
Automorphism Groups ◽  
Measure Algebra ◽  
Automatic Continuity ◽  
Simple Criterion ◽  
Separable Group ◽  
Urysohn Space ◽  
Metric Structures ◽  
Small Index ◽  
Image Position

AbstractWe define a simple criterion for a homogeneous, complete metric structure X that implies that the automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the automatic continuity property, the small index property, and uncountable cofinality for nonopen subgroups. Then we verify it for the Urysohn space $$, the Lebesgue probability measure algebra MALG, and the Hilbert space $\ell _2 $, thus proving that Iso($$), Aut(MALG), $U\left( {\ell _2 } \right)$, and $O\left( {\ell _2 } \right)$ share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group K with a left-invariant, complete metric, is trivial, and we verify it for $$, and $\ell _2 $.

scholarly journals AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

2017 ◽  
Vol 18 (3) ◽  
pp. 561-590 ◽  
Author(s):  
Marcin Sabok
Keyword(s):  
Metric Spaces ◽  
Separable Hilbert Space ◽  
Continuity Property ◽  
Group Topology ◽  
Automatic Continuity ◽  
Separable Group ◽  
Urysohn Space ◽  
Infinite Dimensional ◽  
Group Of Isometries ◽  
Groups Of Isometries

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Induction and Restriction of π-Partial Characters and their Lifts

1996 ◽  
Vol 48 (6) ◽  
pp. 1210-1223 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Subnormal Subgroup ◽  
Primitive Character ◽  
Separable Group ◽  
Ordinary Character ◽  
Brauer Characters ◽  
Alternative Set

AbstractLet G be a finite π-separable group, where π is a set of primes. The π-partial characters of G are the restrictions of the ordinary characters to the set of π-elements of G. Such an object is said to be irreducible if it is not the sum of two nonzero partial characters and the set of irreducible π- partial characters of G is denoted Iπ(G). (If p is a prime and π = p′, then Iπ(G) is exactly the set of irreducible Brauer characters at p.)From their definition, it is obvious that each partial character φ ∊ Iπ(G) can be “lifted” to an ordinary character χ ∊ Irr(G). (This means that φ is the restriction of χ to the π-elements of G.) In fact, there is a known set of canonical lifts Bπ(G) ⊆ Irr(G) for the irreducible π-partial characters. In this paper, it is proved that if 2 ∉ π, then there is an alternative set of canonical lifts (denoted Dπ(G)) that behaves better with respect to character induction.An application of this theory to M-groups is presented. If G is an M-group and S ⊆ G is a subnormal subgroup, consider a primitive character θ ⊆ Irr(S). It was known previously that if |G : S| is odd, then θ must be linear. It is proved here without restriction on the index of S that θ(1) is a power of 2.

On dynamical systems preserving weights

2017 ◽  
Vol 38 (7) ◽  
pp. 2729-2747
Author(s):  
LAVY KOILPITCHAI ◽  
KUNAL MUKHERJEE
Keyword(s):  
Von Neumann Algebra ◽  
Bernoulli Shift ◽  
Ergodic Action ◽  
Spectral Multiplicity ◽  
Weak Mixing ◽  
Locally Compact ◽  
Separable Group ◽  
Von Neumann ◽  
Separable Predual ◽  
Normal Semifinite Trace

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and $\infty$. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift.

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