scholarly journals Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

2016 ◽  
Vol 66 (4) ◽  
pp. 1059-1078
Author(s):  
Lingli Zeng ◽  
Jizhu Nan
Keyword(s):  
Abelian Group ◽  
Unitary Groups ◽  
Group Algebras ◽  
Orthogonal Groups ◽  
Vector Invariant

VECTOR INVARIANT IDEALS OF ABELIAN GROUP ALGEBRAS UNDER THE ACTION OF THE SYMPLECTIC GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350046
Author(s):  
JIZHU NAN ◽  
LINGLI ZENG
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Vector Space ◽  
Group Algebra ◽  
Symplectic Group ◽  
Group Algebras ◽  
Symplectic Groups ◽  
Vector Invariant

Let F be a finite field and let Sp 2ν(F) be the symplectic group over F. If Sp 2ν(F) acts on the F-vector space F2ν, then it can induce an action on the vector space F2ν ⊕ F2ν, defined by (x, y)A = (xA, yA), ∀ x, y ∈ F2ν, A ∈ Sp 2ν(F). If K is a field with char K ≠ char F, then Sp 2ν(F) also acts on the group algebra K[F2ν ⊕ F2ν]. In this paper, we determine the structures of Sp 2ν(F)-stable ideals of the group algebra K[F2ν ⊕ F2ν] by augmentation ideals, and describe the relations between the invariant ideals of K[F2ν] and the vector invariant ideals of K[F2ν ⊕ F2ν].

scholarly journals Existence of finite groups with classical commutator subgroup

1978 ◽  
Vol 25 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Michael D. Miller
Keyword(s):  
Abelian Group ◽  
Finite Fields ◽  
Finite Groups ◽  
Commutator Subgroup ◽  
Unitary Groups ◽  
The Other ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Orthogonal Groups ◽  
General Linear Groups

AbstractGiven a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

WARFIELD INVARIANTS IN COMMUTATIVE GROUP ALGEBRAS

2008 ◽  
Vol 07 (03) ◽  
pp. 337-346 ◽  
Author(s):  
PETER V. DANCHEV
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Group Algebra ◽  
Ordinal Number ◽  
Commutative Group ◽  
Group Algebras

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants Wα, q(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G. This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

scholarly journals Invariant ideals of abelian group algebras under the multiplicative action of a field. II

2001 ◽  
Vol 130 (4) ◽  
pp. 951-957 ◽  
Author(s):  
J. M. Osterburg ◽  
D. S. Passman ◽  
A. E. Zalesskiĭ
Keyword(s):  
Abelian Group ◽  
Group Algebras

Minimal ideals in finite abelian group algebras and coding theory

2016 ◽  
Vol 10 (2) ◽  
pp. 321-340
Author(s):  
Gladys Chalom ◽  
Raul Antonio Ferraz ◽  
Marinês Guerreiro
Keyword(s):  
Abelian Group ◽  
Coding Theory ◽  
Finite Abelian Group ◽  
Group Algebras

Derivations on group algebras of a locally compact abelian group

2015 ◽  
Vol 180 (3) ◽  
pp. 595-605 ◽  
Author(s):  
Mohammad Javad Mehdipour ◽  
Zahra Saeedi
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Group Algebras ◽  
Locally Compact Abelian Group ◽  
Locally Compact

scholarly journals Sur l'homologie des groupes unitaires à coefficients polynomiaux

2012 ◽  
Vol 10 (1) ◽  
pp. 87-139 ◽  
Author(s):  
Aurélien Djament
Keyword(s):  
Unitary Groups ◽  
Symmetric Power ◽  
Orthogonal Groups ◽  
Tensor Power ◽  
Full Generality ◽  
Standard Methods ◽  
Homology Groups ◽  
Special Cases ◽  
Functor Homology ◽  
Kunneth Formula

AbstractLet A be a ring with anti-involution and F a nice functor (tensor or symmetric power, for example) from finitely-generated projective A-modules to abelian groups. We show that the homology of the hyperbolic unitary groups Un,n(A) with coefficients in F(A2n) can be expressed stably (i.e. after taking the colimit over n) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, when A is a field of characteristic 0 or a ring without ℤ-torsion and F a tensor power). This extends the result where A is a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).The proof begins by relating, without any assumption on F, our homology groups to the homology of a category of hermitian spaces with coefficients twisted by F. Then, when F is polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groups On (A) (the ring A being here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it when A is a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.

Complex group algebras of almost simple unitary groups

2020 ◽  
Vol 48 (5) ◽  
pp. 1919-1940
Author(s):  
Farrokh Shirjian ◽  
Ali Iranmanesh ◽  
Farideh Shafiei
Keyword(s):  
Unitary Groups ◽  
Group Algebras ◽  
Complex Group

Sandwich classification for O2n+1(R) and U2n+1(R,Δ) revisited

2018 ◽  
Vol 21 (4) ◽  
pp. 539-571 ◽  
Author(s):  
Raimund Preusser
Keyword(s):  
Natural Number ◽  
Commutative Ring ◽  
Unitary Groups ◽  
Orthogonal Groups

AbstractIn a recent paper, the author proved that if {n\geq 3} is a natural number, R a commutative ring and {\sigma\in GL_{n}(R)}, then {t_{kl}(\sigma_{ij})} where {i\neq j} and {k\neq l} can be expressed as a product of 8 matrices of the form {{}^{\varepsilon}\sigma^{\pm 1}} where {\varepsilon\in E_{n}(R)}. In this article we prove similar results for the odd-dimensional orthogonal groups {O_{2n+1}(R)} and the odd-dimensional unitary groups {U_{2n+1}(R,\Delta)} under the assumption that R is commutative and {n\geq 3}. This yields new, short proofs of the Sandwich Classification Theorems for the groups {O_{2n+1}(R)} and {U_{2n+1}(R,\Delta)}.

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