scholarly journals A nine-dimensional algebra which is not a block of a finite group

2020 ◽  
Author(s):  
Markus Linckelmann ◽  
William Murphy
Keyword(s):  
Finite Group ◽  
Basic Algebra ◽  
Group Algebras ◽  
Closed Field ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Rule Out

Abstract We rule out a certain nine-dimensional algebra over an algebraically closed field to be the basic algebra of a block of a finite group, thereby completing the classification of basic algebras of dimension at most 12 of blocks of finite group algebras.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

scholarly journals On the degree of an indecomposable representation of a finite group

1979 ◽  
Vol 28 (3) ◽  
pp. 321-324 ◽  
Author(s):  
Gerald H. Cliff
Keyword(s):  
Finite Group ◽  
Indecomposable Representation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group

AbstractLet k be an algebraically closed field of characteristic p, and G a finite group. Let M be an indecomposable kG-module with vertex V and source X, and let P be a Sylow p-subgroup of G containing V. Theorem: If dimkX is prime to p and if NG(V) is p-solvable, then the p-part of dimkM equals [P:V]; dimkX is prime to p if V is cyclic.

scholarly journals Block Idempotents and Normal p-Subgroups

1966 ◽  
Vol 28 ◽  
pp. 1-13 ◽  
Author(s):  
W. F. Reynolds
Keyword(s):  
Finite Group ◽  
Reduction Theorem ◽  
Closed Field ◽  
Modular Representations ◽  
Algebraically Closed Field ◽  
A Finite Group

In the theory of modular representations of a finite group G in an algebraically closed field Ω of characteristic p, Brauer has proved a useful reduction theorem for blocks [2, §§11, 12], [5, (88.8)], which can be reformulated as follows:THEOREM 1 (Brauer). Let P bean arbitraryp-subgroup of G; let N = NG(P) and W = PCG(P).

scholarly journals Factor Ideals of Some Representation Algebras

1969 ◽  
Vol 9 (1-2) ◽  
pp. 109-123 ◽  
Author(s):  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Isomorphism Class ◽  
Closed Field ◽  
Complex Field ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Representation Algebra

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).

scholarly journals Steinberg-like characters for finite simple groups

2020 ◽  
Vol 23 (1) ◽  
pp. 25-78
Author(s):  
Gunter Malle ◽  
Alexandre Zalesski
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Regular Representation ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Algebraically Closed Field ◽  
Characteristic 2 ◽  
A Finite Group

AbstractLet G be a finite group and, for a prime p, let S be a Sylow p-subgroup of G. A character χ of G is called {\mathrm{Syl}_{p}}-regular if the restriction of χ to S is the character of the regular representation of S. If, in addition, χ vanishes at all elements of order divisible by p, χ is said to be Steinberg-like. For every finite simple group G, we determine all primes p for which G admits a Steinberg-like character, except for alternating groups in characteristic 2. Moreover, we determine all primes for which G has a projective FG-module of dimension {\lvert S\rvert}, where F is an algebraically closed field of characteristic p.

scholarly journals Isomorphisms of subcategories of fusion systems of blocks and Clifford theory

2020 ◽  
Vol 23 (5) ◽  
pp. 925-930
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
London Math ◽  
Group Algebras ◽  
Closed Field ◽  
Prime Characteristic ◽  
Cambridge University ◽  
Finite Poset ◽  
Clifford Theory ◽  
A Finite Group ◽  
Brauer Pairs

AbstractLet k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a {(G,N,c)}-Brauer pair {(R,f_{R})} consists of a p-subgroup R of G and a block {f_{R}} of {(kC_{N}(R))}. If Q is a defect group of c and {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))}, then {(Q,f_{Q})} is a {(G,N,c)}-Brauer pair. The {(G,N,c)}-Brauer pairs form a (finite) poset. Set {H=N_{G}(Q,f_{Q})} so that {(Q,f_{Q})} is an {(H,C_{N}(Q),f_{Q})}-Brauer pair. We extend Lemma 8.6.4 of the above book to show that if {(U,f_{U})} is a maximal {(G,N,c)}-Brauer pair containing {(Q,f_{Q})}, then {(U,f_{U})} is a maximal {(H,C_{N}(c),f_{Q})}-Brauer pair containing {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of {\mathcal{F}_{(U,f_{U})}(G,N,c)} and {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including {(Q,f_{Q})} and {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.

scholarly journals The centralizer of a subgroup in a group algebra

2012 ◽  
Vol 56 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Susanne Danz ◽  
Harald Ellers ◽  
John Murray
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Closed Field ◽  
List Type ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Centralizer Algebra

AbstractLet F be an algebraically closed field, G be a finite group and H be a subgroup of G. We answer several questions about the centralizer algebra FGH. Among these, we provide examples to show that•the centre Z(FGH) can be larger than the F-algebra generated by Z(FG) and Z(FH),•FGH can have primitive central idempotents that are not of the form ef, where e and f are primitive central idempotents of FG and FH respectively,•it is not always true that the simple FGH-modules are the same as the non-zero FGH-modules HomFH(S, T ↓ H), where S and T are simple FH and FG-modules, respectively.

scholarly journals An Application of Finite Groups to Hopf algebras

2021 ◽  
Vol 14 (3) ◽  
pp. 816-828
Author(s):  
Tahani Al-Mutairi ◽  
Mohammed Mosa Al-shomrani
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Finite Groups ◽  
Hopf Algebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group

Kaplansky’s famous conjectures about generalizing results from groups to Hopf al-gebras inspired many mathematicians to try to find solusions for them. Recently, Cohen and Westreich in [8] and [10] have generalized the concepts of nilpotency and solvability of groups to Hopf algebras under certain conditions and proved interesting results. In this article, we follow their work and give a detailed example by considering a finite group G and an algebraically closed field K. In more details, we construct the group Hopf algebra H = KG and examine its properties to see what of the properties of the original finite group can be carried out in the case of H.

Cliques of Irreducible Representations, Quotient Groups, and Brauer's Theorems on Blocks

1995 ◽  
Vol 47 (5) ◽  
pp. 929-945 ◽  
Author(s):  
Harald Ellers
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
A Finite Group ◽  
Quotient Groups

AbstractAssume k is an algebraically closed field of characteristic p and G is a finite group. If P is a p-subgroup of G such that G = PCG(P), and if H is a normal subgroup of G with P ≤ H, then the number of H-cliques of irreducible k[G]-modules is the same as the number of H/P-cliques of irreducible k[G/P]-modules.

NOETHER SETTINGS FOR CENTRAL EXTENSIONS OF GROUPS WITH ZERO SCHUR MULTIPLIER

2002 ◽  
Vol 01 (01) ◽  
pp. 107-112 ◽  
Author(s):  
ESTHER BENEISH
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cyclic Group ◽  
Central Extension ◽  
Closed Field ◽  
Schur Multiplier ◽  
Central Extensions ◽  
Algebraically Closed Field ◽  
Rational Extension ◽  
A Finite Group

Let G be a finite group, and let F be an algebraically closed field. The rational extension of F, F(G) = F(xg : g ∈ G) with G action given by hxg = xhg for g, h ∈ G, is referred to as the Noether setting for G. We show that if G is a group with zero Schur multiplier, then for any central extension G′ of G, F(G′)G′ and F(G)G are stably equivalent over F. That is, F(G)G is stably rational over F if and only if F(G′)G′ is stably rational over F. In particular, if G is a cyclic group or an abelian group with zero Schur multiplier, then F(G′) is stably rational over F.

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