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.

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.

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.

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.

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.

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.

On Extending Projectives of Finite Group-Graded Algebras

1991 ◽  
Vol 34 (2) ◽  
pp. 224-228
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Group Representation ◽  
Sufficient Conditions ◽  
Projective Cover ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

Free Subgroups of Units in Group Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 309-312 ◽  
Author(s):  
Jairo Zacarias Gonçalves
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Free Subgroup ◽  
Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Free Subgroups

AbstractIn this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.Some extensions of this results to infinite nilpotent and FC groups are also considered.

scholarly journals PRIMITIVE CENTRAL IDEMPOTENTS OF RATIONAL GROUP ALGEBRAS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250130
Author(s):  
GEOFFREY JANSSENS
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Group Algebras ◽  
Rational Group ◽  
A Finite Group ◽  
Del Rio

We give a description of the primitive central idempotents of the rational group algebra ℚG of a finite group G. Such a description is already investigated by Jespers, Olteanu and del Río, but some unknown scalars are involved. Our description also gives answers to their questions.

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