THE RATIONAL GROUP ALGEBRA OF A FINITE GROUP

2012 ◽  
Vol 12 (03) ◽  
pp. 1250168 ◽  
Author(s):  
GURMEET K. BAKSHI ◽  
RAVINDRA S. KULKARNI ◽  
INDER BIR S. PASSI
Keyword(s):  
Finite Group ◽  
Explicit Expression ◽  
Group Algebra ◽  
Irreducible Character ◽  
Metabelian Group ◽  
Central Idempotent ◽  
Rational Group ◽  
Wedderburn Decomposition ◽  
Complete Set ◽  
A Finite Group

An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] of a finite group G associated with any complex irreducible character of G is obtained. A complete set of primitive central idempotents and the Wedderburn decomposition of the rational group algebra of a finite metabelian group is also computed.

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 On a theorem of J. A. Green

1975 ◽  
Vol 20 (4) ◽  
pp. 449-450 ◽  
Author(s):  
Michael F. O'Reilly
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Short Proof ◽  
Central Idempotent ◽  
A Finite Group

Let G be a finite group, K a field of characteristic p and Γ the group algebra of G over k. Let e = ∑g∈Gαgg, αg ∈k, be a primitive central idempotent of Γ; let supp e = {g ∈ G: αg≠0}. We provide a short proof of a slightly stronger version of Theorem 5 of Green (1968).

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

The largest Irreducible Character Degree of a Finite Group

1985 ◽  
Vol 37 (3) ◽  
pp. 442-451 ◽  
Author(s):  
David Gluck
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Abelian Subgroup ◽  
Character Degree ◽  
Character Table ◽  
Significant Information ◽  
Famous Theorem ◽  
Frobenius Reciprocity ◽  
A Finite Group ◽  
Irreducible Character Degree

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

scholarly journals A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

scholarly journals Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

2019 ◽  
Vol 22 (5) ◽  
pp. 953-974
Author(s):  
Ángel del Río ◽  
Mariano Serrano
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Infinite Family ◽  
Solvable Groups ◽  
Integral Group ◽  
Rational Group ◽  
Torsion Unit ◽  
Zassenhaus Conjecture ◽  
A Finite Group ◽  
Torsion Units

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.

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.

scholarly journals On characters in the principal 2-block

1978 ◽  
Vol 25 (3) ◽  
pp. 264-268 ◽  
Author(s):  
Thomas R. Berger ◽  
Marcel Herzog
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cyclic Group ◽  
Complex Number ◽  
Irreducible Character ◽  
Odd Order ◽  
A Finite Group

AbstractLet k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

Sign in / Sign up

Export Citation Format

Share Document