scholarly journals The semicentre of a group algebra

1999 ◽  
Vol 42 (1) ◽  
pp. 95-111 ◽  
Author(s):  
Paul Wauters
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Element ◽  
Necessary And Sufficient ◽  
Torsion Free

We study the semicentre of a group algebra K[G] where K is a field of characteristic zero and G is a polycyclic-by-finite group suchthat Δ(G) is torsion-free abelian. Several properties about the structure of this ring are proved, in particular as to when is the semicentre a UFD. Examples are constructed when this is not the case. We also prove necessary and sufficient conditions for every normal element of K[G] which belongs to K[Δ(G)] to be the product of a unit and a semi-invariant.

GROUP ALGEBRAS WITH UNIT GROUPS OF DERIVED LENGTH THREE

2010 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Characteristic P ◽  
Derived Length ◽  
A Finite Group ◽  
Necessary And Sufficient

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained.

scholarly journals Group Algebras With Central Radicals

1962 ◽  
Vol 5 (3) ◽  
pp. 103-108 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
A Finite Group ◽  
Necessary And Sufficient

It is well known that when the characteristic p(≠ 0) of a field divides the order of a finite group, the group algebra possesses a non-trivial radical and that, if p does not divide the order of the group, the group algebra is semi-simple. A group algebra has a centre, a basis for which consists of the class-sums. The radical may be contained in this centre; we obtain necessary and sufficient conditions for this to happen.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

Group Algebras with Minimal Strong Lie Derived Length

2008 ◽  
Vol 51 (2) ◽  
pp. 291-297 ◽  
Author(s):  
Ernesto Spinelli
Keyword(s):  
Solvable Group ◽  
Group Algebra ◽  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Derived Length ◽  
Necessary And Sufficient

AbstractLet KG be a non-commutative strongly Lie solvable group algebra of a group G over a field K of positive characteristic p. In this note we state necessary and sufficient conditions so that the strong Lie derived length of KG assumes its minimal value, namely [log2(p + 1)].

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group

1970 ◽  
Vol 22 (1) ◽  
pp. 41-46 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Proper Subgroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fitting Subgroup ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.

CONSTRUCTION AND PROPERTIES OF QUASI RICCI FLAT SPACES

1986 ◽  
Vol 01 (04) ◽  
pp. 997-1007 ◽  
Author(s):  
GUY BONNEAU ◽  
FRANÇOIS DELDUC
Keyword(s):  
String Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Real ◽  
Ricci Flat ◽  
Compact Case ◽  
Flat Manifolds ◽  
Necessary And Sufficient ◽  
Lagrangian Densities ◽  
Torsion Free

We look for the necessary and sufficient conditions for a generalized torsion-free nonlinear σ-model to be one-loop finite. The corresponding metrics are not only Ricci flat ones, but also a larger class we call “quasi Ricci flat” spaces. We give expressions for the corresponding Lagrangian densities in the real and Kähler cases. In the latter, the manifold is shown to be proper, complete and nonhomogeneous. Unfortunately, in the compact case, relevant for string theory, these quasi Ricci flat manifolds become Ricci flat ones.

scholarly journals Two remarks on amalgams of locally finite groups

1987 ◽  
Vol 36 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Berthold J. Maier
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
The Common ◽  
Necessary And Sufficient

We construct non amalgamation bases in the class of locally finite groups, and we present necessary and sufficient conditions for the embeddability of an amalgam into a locally finite group in the case that the common subgroup has finite index in both constituents.

A complete characterization of the existence of rational functional solutions with infinite support bases

2016 ◽  
Vol 15 (09) ◽  
pp. 1650178 ◽  
Author(s):  
Lan Nguyen
Keyword(s):  
Rational Function ◽  
Characteristic Zero ◽  
Polynomial Solution ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Solutions ◽  
Function Solution ◽  
Necessary And Sufficient

It is known that there exist polynomial solutions [Formula: see text] with infinite support base [Formula: see text], of certain functional equations arising from quantum arithmetics, which cannot be constructed from quantum integers. A description of the necessary and sufficient conditions on a set of primes [Formula: see text] for the existence of a polynomial solution, with field of coefficients of characteristic zero and support base [Formula: see text], which cannot be constructed from quantum integers is also known, leading to the classification of the set of polynomial solutions. In his papers on quantum arithmetics, Melvyn Nathanson raises a question concerning the classification of the possibly non-trivially broader set of solutions, namely the set of rational function solutions. It is not known at the time that the set of rational function solutions is more than just the set of ratio of polynomial solutions. However, it is now known that there are infinitely many rational function solutions [Formula: see text], with support base [Formula: see text] and field of coefficients of characteristic zero, which are not ratios of polynomial solutions with the same support base, even in the purely cyclotomic case. Thus, a natural question that should be asked in order to classify the set of rational function solutions, is: If polynomial solutions are replaced by merely rational function solutions, what would the necessary and sufficient conditions be on the support base [Formula: see text]? In this paper, we give a complete description of the necessary and sufficient conditions on the set of primes [Formula: see text] for the existence of a rational function solution, with field of coefficients of characteristic zero and support base [Formula: see text], which cannot be constructed from quantum integers.

On c#-normal subgroups of finite groups

2016 ◽  
Vol 16 (08) ◽  
pp. 1750160
Author(s):  
Guo Zhong ◽  
Shi-Xun Lin
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Necessary And Sufficient

Let [Formula: see text] be a subgroup of a finite group [Formula: see text]. We say that [Formula: see text] is a [Formula: see text]-normal subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a [Formula: see text]-subgroup of [Formula: see text]. In the present paper, we use [Formula: see text]-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be [Formula: see text]-supersolvable, [Formula: see text]-nilpotent and solvable. Our results extend and improve some recent ones.

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