Group algebras with radicals of square zero

1962 ◽  
Vol 5 (4) ◽  
pp. 158-159 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Group Algebras ◽  
A Finite Group ◽  
Special Case

Over a field of characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself.

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 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.

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.

scholarly journals Twisted group algebras and their representations

1964 ◽  
Vol 4 (2) ◽  
pp. 152-173 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Group Algebras ◽  
Linear Combinations ◽  
Twisted Group Algebra ◽  
Formula One ◽  
Twisted Group ◽  
Image Position ◽  
A Finite Group ◽  
Group Product

Let be a finite group, a field. A twisted group algebra A() on over is an associative algebra whose elements are the formal linear combinations and in which the product (A)(B) is a non-zero multiple of (AB), where AB is the group product of A, B ∈: . One gets the ordinary group algebra () by taking each fA, B ≠ 1.

scholarly journals Remarks on the commutativity of the radicals of group algebras

1979 ◽  
Vol 20 (1) ◽  
pp. 63-68 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Commutator Subgroup ◽  
Group Algebras ◽  
Arbitrary Field ◽  
A Finite Group

Let K be an arbitrary field with characteristic p > 0, G a finite group of order pag′ with (p, g′) = 1, P a p-Sylow subgroup of G and G′ the commutator subgroup of G. For a ring R denote by J(R) the Jacobson radical of R and by Z(R) the centre of R. We write KG for the group algebra of G over K.

Varieties and modules with vanishing cohomology

1994 ◽  
Vol 116 (2) ◽  
pp. 245-251 ◽  
Author(s):  
Jon F. Carlson ◽  
Geoffrey R. Robinson
Keyword(s):  
Finite Group ◽  
Projective Module ◽  
Group Algebras ◽  
Finite Characteristic ◽  
Principal Block ◽  
Vanishing Of Cohomology ◽  
A Finite Group ◽  
Special Case

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context of kG-modules where k is a field of finite characteristic p and G is a finite group whose order is divisible by p. Aside from some general considerations, the main results of [2] related the existence of kG-modules M with H*(G, M) = 0 to the structure of the centralizers of the p-elements in G. Specifically it was shown that there exists a non-projective module M in the principal block of kG with H*(G, M) = 0 whenever the centralizer of some p-element of G is not p-nilpotent. The converse was proved in the special case that the prime p is an odd integer (p > 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.

On strongly associative group algebras of p-solvable groups

2015 ◽  
Vol 14 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Jonas Gonçalves Lopes
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Group Algebra ◽  
Modular Group ◽  
Solvable Groups ◽  
Crossed Product ◽  
Group Algebras ◽  
Characteristic P ◽  
Partial Action ◽  
A Finite Group

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH *α G. If FH *α G is associative for any G and any α, then FH is called strongly associative. We characterize the strongly associative modular group algebras FH with H being a p-solvable group.

scholarly journals Excellent Extensions and Homological Conjectures

2018 ◽  
Vol 25 (04) ◽  
pp. 619-626
Author(s):  
Yingying Zhang
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
A Finite Group ◽  
Excellent Extension ◽  
Special Case ◽  
Skew Group Algebra ◽  
Dimension Conjecture ◽  
Homological Conjectures

In this paper, we introduce the notion of excellent extensions of rings. Let Γ be an excellent extension of an Artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp., finitistic dimension conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, when G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebra ΛG satisfies strong Nakayama conjecture (resp., generalized Nakayama conjecture), then so does Λ.

scholarly journals Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Constantin-Cosmin Todea
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Finite Groups ◽  
Hochschild Cohomology ◽  
Product Formula ◽  
Group Algebras ◽  
Closed Field ◽  
Additive Decomposition ◽  
A Finite Group ◽  
Block Algebra

AbstractWe give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a product formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For an algebraically closed field

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

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