scholarly journals Algebraic construction of quasi-split algebraic tori

2019 ◽  
Vol 19 (11) ◽  
pp. 2050206
Author(s):  
Armin Jamshidpey ◽  
Nicole Lemire ◽  
Éric Schost
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Explicit Construction ◽  
Constructive Proof ◽  
Algebraic Construction ◽  
Quotient Field ◽  
Algebraic Tori ◽  
Transcendental Extension ◽  
A Finite Group

The main purpose of this work is to give a constructive proof for a particular case of the no-name lemma. Let [Formula: see text] be a finite group, [Formula: see text] a field that is equipped with a faithful [Formula: see text]-action, and [Formula: see text] a sign permutation [Formula: see text]-lattice (see the Introduction for the definition). Then [Formula: see text] acts naturally on the group algebra [Formula: see text] of [Formula: see text] over [Formula: see text], and hence also on the quotient field [Formula: see text]. A well-known variant of the no-name lemma asserts that the invariant sub-field [Formula: see text] is a purely transcendental extension of [Formula: see text]. In other words, there exist [Formula: see text] which are algebraically independent over [Formula: see text] such that [Formula: see text]. In this paper, we give an explicit construction of suitable elements [Formula: see text].

scholarly journals Group structure and properties of block ideals of the group algebra

1975 ◽  
Vol 16 (1) ◽  
pp. 22-28 ◽  
Author(s):  
Wolfgang Hamernik
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Group Structure ◽  
Arbitrary Field ◽  
Structure And Properties ◽  
Characteristic P ◽  
Principal Block ◽  
A Finite Group

In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:(i) J(B1) ≤ Z(B1)(ii) J(B1)is commutative,(iii) G is p-nilpotent with abelian Sylowp-subgroups.

Invariants of hyperplane groups and vanishing ideals of finite sets of points

2012 ◽  
Vol 55 (2) ◽  
pp. 355-367 ◽  
Author(s):  
H. E. A. Campbell ◽  
Jianjun Chuai
Keyword(s):  
Finite Group ◽  
Polynomial Ring ◽  
Constructive Proof ◽  
Additive Subgroup ◽  
Finite Sets ◽  
Vanishing Ideal ◽  
Invariant Ring ◽  
Invariant Differential ◽  
A Finite Group ◽  
Vanishing Ideals

AbstractWe define a hyperplane group to be a finite group generated by reflections fixing a single hyperplane pointwise. Landweber and Stong proved that the invariant ring of a hyperplane group is again a polynomial ring in any characteristic. Recently, Hartmann and Shepler gave a constructive proof of this result. By their algorithm, one can always construct generators that are additive. In this paper, we study hyperplane groups of order a power of a prime p in characteristic p and give a slightly different construction of the generators than Hartmann and Shepler. We then show that such generators have a particular form. Furthermore, we show that if the group is defined by a finite additive subgroup W ⊆ $W\subseteq\mathbb{F}^n$, the vanishing ideal of W is generated by polynomials obtained from a set of generators of the invariant ring that are additive. Finally, we give a shorter proof of the fact that the module of the invariant differential 1-forms is free in our situation.

scholarly journals A characterization of PGL (2,pn) by some irreducible complex character degrees

2016 ◽  
Vol 99 (113) ◽  
pp. 257-264 ◽  
Author(s):  
Somayeh Heydari ◽  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Group Algebra ◽  
Character Degrees ◽  
Complex Character ◽  
Complex Group ◽  
Complex Group Algebra ◽  
A Finite Group

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2,p) |, p ? cd(G) and max(cd(G)) = p+1, then G ? PGL(2,p), SL(2, p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also, we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G ? PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely determined by the structure of its complex group algebra.

scholarly journals A note on vertices of indecomposable tensor products

2020 ◽  
Vol 23 (3) ◽  
pp. 385-391
Author(s):  
Markus Linckelmann
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Present Note ◽  
Tensor Products ◽  
Solvable Groups ◽  
Sufficient Criterion ◽  
Indecomposable Modules ◽  
Main Motivation ◽  
Simple Modules ◽  
A Finite Group

AbstractG. Navarro raised the question of when two vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup. The present note provides a sufficient criterion for this to happen. This generalises a result by Navarro for simple modules over finite p-solvable groups, which is the main motivation for this note.

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 On the commutativity of the radical of a group algebra

1965 ◽  
Vol 7 (1) ◽  
pp. 1-8 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Prime Characteristic ◽  
Different Types ◽  
A Finite Group

Over a field of prime 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. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative.

Sign in / Sign up

Export Citation Format

Share Document