A family of crystallographic groups with 2-torsion in K0 of the rational group algebra

1991 ◽  
Vol 34 (2) ◽  
pp. 325-331 ◽  
Author(s):  
P. H. Kropholler ◽  
B. Moselle
Keyword(s):  
Group Algebra ◽  
Euler Class ◽  
Crystallographic Group ◽  
Rational Group ◽  
Crystallographic Groups

We calculate K0 of the rational group algebra of a certain crystallographic group, showing that it contains an element of order 2. We show that this element is the Euler class, and use our calculation to produce a whole family of groups with Euler class of order 2.

On the Garsia Lie Idempotent

2005 ◽  
Vol 48 (3) ◽  
pp. 445-454 ◽  
Author(s):  
Frédéric Patras ◽  
Christophe Reutenauer ◽  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Group Algebra ◽  
Orthogonal Projection ◽  
Gram Matrix ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Rational Group ◽  
Lie Idempotent

AbstractThe orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group Sn, in each homogenous degree n. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of Sn−1. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in Sn.

Central Idempotents and Units in Rational Group Algebras of Alternating Groups

1998 ◽  
Vol 08 (04) ◽  
pp. 467-477 ◽  
Author(s):  
A. Giambruno ◽  
E. Jespers
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Explicit Description ◽  
Alternating Group ◽  
Group Algebras ◽  
Young Tableaux ◽  
Rational Group ◽  
Alternating Groups ◽  
Group Of Units

Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.

Invertible powers of ideals over orders in commutative separable alǵebras

1970 ◽  
Vol 67 (2) ◽  
pp. 237-242 ◽  
Author(s):  
Michael Singer
Keyword(s):  
Group Algebra ◽  
Sylow Subgroup ◽  
Basic Result ◽  
Rational Group ◽  
Powers Of Ideals ◽  
Dedekind Domains ◽  
Result Theorem ◽  
Quantitative Results ◽  
Separable Algebras ◽  
Special Case

The purpose of this paper is to obtain quantitative results on invertible powers of (fractional) ideals in commutative separable algebras over dedekind domains. This is connected with the work of Dade, Taussky and Zassenhaus(2) on ideals in noetherian domains. We do not, however, make use of their paper, but rather draw on the general theorems on ideals in commutative separable algebras established by Fröhlich (3), in particular his qualitative result that some power of any given ideal is invertible. Our basic result (Theorem 1) concerns the invertibility of powers of a particular type of ideal, the componentwise dedekind ideals defined below. From this we deduce a general result (Theorem 2), which includes as a special case Theorem C of (2) for the case of separable field extensions; specifically, the (n – 1)th power of any ideal is invertible, where n is the dimension of the algebra. Although, as we show, it is possible to deduce Theorem 2 from (2), we have here an independent proof of one of the main results of (2) based entirely on the results in (3). As a further application of Theorem 1 we obtain a new result on ideals over the group ring of an abelian group over the ring of rational integers; the (t – 1)th power of such an ideal is invertible, where t is the maximum number of simple components of the rational group algebra of any Sylow subgroup. We also show that this is the best possible result when some Sylow subgroup whose rational group algebra has t components is cyclic.

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.

The Group of Automorphisms of the Rational Group Algebra of a Finite Metacyclic Group

2006 ◽  
Vol 34 (10) ◽  
pp. 3543-3567 ◽  
Author(s):  
Aurora Olivieri ◽  
Á. del Río ◽  
Juan Jacobo Simón
Keyword(s):  
Group Algebra ◽  
Metacyclic Group ◽  
Rational Group ◽  
Group Of Automorphisms

scholarly journals The rational group algebra of a normally monomial group

2014 ◽  
Vol 218 (9) ◽  
pp. 1583-1593 ◽  
Author(s):  
Gurmeet K. Bakshi ◽  
Sugandha Maheshwary
Keyword(s):  
Group Algebra ◽  
Rational Group ◽  
Monomial Group

The R∞ property for crystallographic groups of Sol14

2019 ◽  
Vol 19 (04) ◽  
pp. 2050072
Author(s):  
Won Sok Yoo
Keyword(s):  
Crystallographic Group ◽  
Crystallographic Groups ◽  
Reidemeister Number

For any automorphism of a crystallographic group of [Formula: see text], we prove that its Reidemeister number is infinite.

scholarly journals Crystallographic groups with trivial center and outer automorphism group

2017 ◽  
Vol 164 (2) ◽  
pp. 363-368
Author(s):  
RAFAŁ LUTOWSKI ◽  
ANDRZEJ SZCZEPAŃSKI
Keyword(s):  
Automorphism Group ◽  
Outer Automorphism ◽  
Crystallographic Group ◽  
Crystallographic Groups ◽  
Outer Automorphism Group ◽  
Trivial Center ◽  
Cocompact Subgroup ◽  
Discrete Cocompact Subgroup

AbstractLet Γ be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(ℝn) = O(n) ⋉ ℝn. For any n ⩾ 2, we construct a crystallographic group with a trivial center and trivial outer automorphism group.

On Groups of Order p3

1963 ◽  
Vol 15 ◽  
pp. 622-624 ◽  
Author(s):  
James A. Cohn ◽  
Donald Livingstone
Keyword(s):  
Group Algebra ◽  
Abelian Groups ◽  
Rational Numbers ◽  
The Other ◽  
Rational Group ◽  
Character Tables ◽  
The One

The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.

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