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.

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 $G$.

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.

LIE SOLVABLE GROUP ALGEBRAS OF DERIVED LENGTH THREE IN CHARACTERISTIC THREE

2012 ◽  
Vol 11 (05) ◽  
pp. 1250098 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Solvable Group ◽  
Commutator Subgroup ◽  
Group Algebras ◽  
Derived Length ◽  
Engel Group

In this paper we provide a characterization of Lie solvable group algebras of derived length three over a field of characteristic three when G is a non-2-Engel group with abelian commutator subgroup.

The derived length of the unit group of a group algebra — The case G′ = Sylp(G)

2016 ◽  
Vol 16 (08) ◽  
pp. 1750142 ◽  
Author(s):  
Tibor Juhász
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Derived Length ◽  
Group Of Units

Let [Formula: see text] be an odd prime, and let [Formula: see text] be a nilpotent group, whose commutator subgroup is finite abelian satisfying [Formula: see text] and [Formula: see text]. In this contribution, an upper bound is given on the derived length of the group of units of the group algebra of [Formula: see text] over a field of characteristic [Formula: see text]. Furthermore, we show that this bound is achieved, whenever [Formula: see text] is cyclic.

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.

scholarly journals Group algebras whose p-elements form a subgroup

2016 ◽  
Vol 16 (09) ◽  
pp. 1750170
Author(s):  
M. Ramezan-Nassab
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Group Algebra ◽  
Polynomial Identity ◽  
Matrix Polynomial ◽  
Unit Group ◽  
Group Algebras ◽  
Locally Finite ◽  
P Elements

Let [Formula: see text] be a group, [Formula: see text] a field of characteristic [Formula: see text], and [Formula: see text] the unit group of the group algebra [Formula: see text]. In this paper, among other results, we show that if either (1) [Formula: see text] satisfies a non-matrix polynomial identity, or (2) [Formula: see text] is locally finite, [Formula: see text] is infinite and [Formula: see text] is an Engel-by-finite group, then the [Formula: see text]-elements of [Formula: see text] form a (normal) subgroup [Formula: see text] and [Formula: see text] is abelian (here, of course, [Formula: see text] if [Formula: see text]).

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.

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.

scholarly journals Lie Nilpotency Indices of Modular Group Algebras

2010 ◽  
Vol 17 (01) ◽  
pp. 17-26 ◽  
Author(s):  
V. Bovdi ◽  
J. B. Srivastava
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Commutator Subgroup ◽  
Positive Characteristic ◽  
Group Algebras ◽  
Characteristic P ◽  
Nilpotency Index ◽  
Field Of Positive Characteristic

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G′| + 1, where |G′| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal has already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.

scholarly journals ON FOX QUOTIENTS OF ARBITRARY GROUP ALGEBRAS

2010 ◽  
Vol 20 (05) ◽  
pp. 619-660 ◽  
Author(s):  
MANFRED HARTL
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Abelian Groups ◽  
Group Algebras ◽  
Augmentation Ideal ◽  
Arbitrary Group ◽  
Lie Rings

Certain subquotients of group algebras are determined as a basis for subsequent computations of relative Fox and dimension subgroups. More precisely, for a group G and N-series [Formula: see text] of G let [Formula: see text], n ≥ 0, denote the filtration of the group algebra R(G) induced by [Formula: see text], and IR(G) its augmentation ideal. For subgroups H of G, left ideals J of R(H) and right H-submodules M of [Formula: see text] the quotients IR(G)J/MJ are studied by homological methods, notably for M = IR(G)IR(H), IR(H)IR(G) + I([H, G])R(G) and [Formula: see text] for a normal subgroup N in G; in the latter case the module IR(G)J/MJ is completely determined for n = 2. The groups [Formula: see text] are studied and explicitly computed for n ≤ 3 in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups.

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