scholarly journals The local dimension of a finite group over a number field

2022 ◽  
Author(s):  
Joachim König ◽  
Daniel Neftin
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Local Dimension ◽  
A Finite Group

scholarly journals On the Block of Defect Zero

1971 ◽  
Vol 44 ◽  
pp. 57-59 ◽  
Author(s):  
Yukio Tsushima
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Prime Divisor ◽  
Residue Class ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Residue Class Field ◽  
A Finite Group

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

Restricted Determinantal Homomorphisms and Locally Free Class Groups

1990 ◽  
Vol 42 (4) ◽  
pp. 646-658 ◽  
Author(s):  
Victor Snaith
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Number Field ◽  
Class Group ◽  
Class Groups ◽  
A Finite Group

Let K be a number field and let OK denote the integers of K. The locally free class groups, Cl(OK[G]), furnish a fundamental collection of invariants of a finite group, G. In this paper I will construct some new, non-trivial homomorphisms, called restricted determinants, which map the NGH-invariant idèlic units of Ok([Hab] to Cl(OK[G]). These homomorphisms are constructed by means of the Horn-description of Cl(OK[G]), which describes the locally free class group in terms of the representation theory of G, and the technique of Explicit Brauer Induction, which was introduced in [5].

Equivariant Witt Groups

1990 ◽  
Vol 33 (2) ◽  
pp. 207-218
Author(s):  
Jorge F. Morales
Keyword(s):  
Finite Group ◽  
Number Field ◽  
A Finite Group ◽  
Witt Groups

AbstractThis paper studies for a number field K and a finite group Γ the cokernel of the residue homomorphism .

On the Structure of Group Algebras, I

1965 ◽  
Vol 17 ◽  
pp. 583-593 ◽  
Author(s):  
James A. Cohn ◽  
Donald Livingstone
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Special Class ◽  
Algebraic Number Field ◽  
Group Algebras ◽  
Basic Question ◽  
List Type ◽  
Ring Of Algebraic Integers ◽  
A Finite Group

With this paper we begin a study of the structure of the group algebra RG of a finite group G over the ring of algebraic integers R in an algebraic number field k. The basic question is whether non-isomorphic groups can have isomorphic algebras over R. We shall show that this is impossible if G is (a) abelian,(b) Hamiltonian,(c) one of a special class of p-groups.

scholarly journals Bott Periodicity for group rings An Appendix to “Periodicity of Hermitian K-groups”

2011 ◽  
Vol 7 (3) ◽  
pp. 495-498 ◽  
Author(s):  
Charles Weibel
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Group Rings ◽  
Bott Periodicity ◽  
A Finite Group

AbstractWe show that the groups Kn(RG;ℤ/m) are Bott-periodic for n ≥ 1 whenever G is a finite group, m is prime to |G|, R is a ring of S-integers in a number field and 1/m ∊ R.

A Generalized Brauer Induction Theorem and Its Converse

2012 ◽  
Vol 19 (03) ◽  
pp. 427-432 ◽  
Author(s):  
Gang Chen
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Complex Number ◽  
Prime Numbers ◽  
Linear Map ◽  
The Family ◽  
A Finite Group ◽  
Complex Number Field ◽  
Character Ring

Let G be a finite group, S a unitary subring of the complex number field ℂ and R(G) the character ring of G. Let π be the set of rational prime numbers whose inverses do not belong to S. Denote the family of all p-elementary subgroups of G by W(π), where p runs over π. It is proved that, in the sense of conjugation, W(π) is the least family [Formula: see text] of subgroups of G such that the S-linear map [Formula: see text] is surjective.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

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