scholarly journals GROUP ALGEBRAS WHOSE GROUP OF UNITS IS POWERFUL

2009 ◽  
Vol 87 (3) ◽  
pp. 325-328
Author(s):  
VICTOR BOVDI
Keyword(s):  
Finite Field ◽  
Group Algebra ◽  
Group Algebras ◽  
Group Of Units ◽  
Normalized Units

AbstractA p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

On *-clean non-commutative group rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650150 ◽  
Author(s):  
Hongdi Huang ◽  
Yuanlin Li ◽  
Gaohua Tang
Keyword(s):  
Finite Field ◽  
Local Ring ◽  
Group Algebra ◽  
Rational Numbers ◽  
Semisimple Group ◽  
Group Algebras ◽  
Group Rings ◽  
Dihedral Groups ◽  
Commutative Local Ring ◽  
Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

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.

Twisted Group Rings Whose Units Form an FC-Group

1995 ◽  
Vol 47 (2) ◽  
pp. 274-289
Author(s):  
Victor Bovdi
Keyword(s):  
Group Algebra ◽  
Group Algebras ◽  
Group Rings ◽  
Twisted Group Algebra ◽  
Group Of Units ◽  
Twisted Group

AbstractLet U(KλG) be the group of units of the infinite twisted group algebra KλG over a field K. We describe the FC-centre ΔU of U(KλG) and give a characterization of the groups G and fields K for which U(KλG) = ΔU. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus theorem.

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 On the orders of conjugacy classes in group algebras of p-groups

2004 ◽  
Vol 77 (2) ◽  
pp. 185-190 ◽  
Author(s):  
A. Bovdi ◽  
L. G. Kovács ◽  
S. Mihovski
Keyword(s):  
Conjugacy Class ◽  
Group Algebra ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Normalized Units

AbstractLet p be a prime, a field of pn elements, and G a finite p-group. It is shown here that if G has a quotient whose commutator subgroup is of order p and whose centre has index pk, then the group of normalized units in the group algebra has a conjugacy class of pnk elements. This was first proved by A. Bovdi and C. Polcino Milies for the case k = 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.

GROUP ALGEBRAS WHOSE UNIT GROUP IS LOCALLY NILPOTENT

2020 ◽  
Vol 109 (1) ◽  
pp. 17-23 ◽  
Author(s):  
V. BOVDI
Keyword(s):  
Group Algebra ◽  
Unit Group ◽  
Complete List ◽  
Group Algebras ◽  
Engel Group ◽  
Nilpotent Elements ◽  
Locally Nilpotent ◽  
Normalized Units

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units $V(FG)$ of the group algebra $FG$ is locally nilpotent; (ii) the set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

GROUP ALGEBRAS WITH ENGEL UNIT GROUPS

2016 ◽  
Vol 101 (2) ◽  
pp. 244-252 ◽  
Author(s):  
M. RAMEZAN-NASSAB
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Group Algebras ◽  
Abelian Normal Subgroup ◽  
Locally Finite ◽  
Engel Group ◽  
Group Of Units ◽  
Nilpotent Elements ◽  
Locally Nilpotent ◽  
Formula Group

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$.

scholarly journals Group algebras with an Engel group of units

2006 ◽  
Vol 80 (2) ◽  
pp. 173-178 ◽  
Author(s):  
A. Bovdi
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Commutator Subgroup ◽  
Factor Group ◽  
Group Algebras ◽  
Engel Group ◽  
Group Of Units

AbstractLet F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

Unit groups of finite group algebras of Abelian groups of order at most 16

2020 ◽  
pp. 2150030
Author(s):  
Meena Sahai ◽  
Sheere Farhat Ansari
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Field ◽  
Group Algebra ◽  
Abelian Groups ◽  
Unit Group ◽  
Group Algebras

In this paper, we establish the structure of the unit group of the group algebra [Formula: see text] where [Formula: see text] is an abelian group of order at most 16 and [Formula: see text] is a finite field of characteristic [Formula: see text] with [Formula: see text] elements.

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