Strongly Lie nilpotent group algebras of index at most 8

2014 ◽  
Vol 13 (07) ◽  
pp. 1450044 ◽  
Author(s):  
Harish Chandra ◽  
Meena Sahai
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Characteristic P ◽  
Arbitrary Group

Let K be a field of characteristic p > 0 and let G be an arbitrary group. In this paper, we classify group algebras KG which are strongly Lie nilpotent of index at most 8. We also show that for k ≤ 6, KG is strongly Lie nilpotent of index k if and only if it is Lie nilpotent of index k.

A note on Lie nilpotent group algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950163
Author(s):  
Meena Sahai ◽  
Bhagwat Sharan
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Finite Abelian Group ◽  
Index Formula ◽  
Group Algebras ◽  
Arbitrary Group ◽  
Nilpotency Index

Let [Formula: see text] be an arbitrary group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we give some improvements of the upper bound of the lower Lie nilpotency index [Formula: see text] of the group algebra [Formula: see text]. We also give improved bounds for [Formula: see text], where [Formula: see text] is the number of independent generators of the finite abelian group [Formula: see text]. Furthermore, we give a description of the Lie nilpotent group algebra [Formula: see text] with [Formula: see text] or [Formula: see text]. We also show that for [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the upper Lie nilpotency index of [Formula: see text].

COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350044
Author(s):  
TIBOR JUHÁSZ ◽  
ENIKŐ TÓTH
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Group Algebras ◽  
Augmentation Ideal ◽  
Characteristic P ◽  
Derived Length ◽  
Commutator Identity ◽  
Nilpotency Index ◽  
Powerful Group

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

Tits polygons

2022 ◽  
Vol 275 (1352) ◽  
Author(s):  
Bernhard Mühlherr ◽  
Richard Weiss ◽  
Holger Petersson
Keyword(s):  
Rational Points ◽  
Jordan Algebras ◽  
Parabolic Subgroups ◽  
Characteristic P ◽  
Arbitrary Group ◽  
Content Type ◽  
Defining Relations ◽  
Moufang Condition ◽  
Unique Vertex ◽  
Natural Way

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank  2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least  4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank  1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Lie Dimension Subgroups and Central Series Related to Group Algebras

2009 ◽  
Vol 16 (03) ◽  
pp. 427-436
Author(s):  
Ernesto Spinelli
Keyword(s):  
Group Algebra ◽  
Positive Characteristic ◽  
Nilpotency Class ◽  
Group Algebras ◽  
Central Series ◽  
Characteristic P

Let KG be the group algebra of a group G over a field K of positive characteristic p, and let 𝔇(n)(G) and 𝔇[n](G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality 𝔇(n)(G) =𝔇[n](G) is verified when p ≥ 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p ≤ 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG.

GROUP ALGEBRAS WITH UNIT GROUPS OF DERIVED LENGTH THREE

2010 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Characteristic P ◽  
Derived Length ◽  
A Finite Group ◽  
Necessary And Sufficient

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained.

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.

On Unit Groups in Modular Group Algebras of Abelian Groups with Totally Projective Components

2007 ◽  
Vol 14 (03) ◽  
pp. 515-520
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Modular Group ◽  
Abelian Groups ◽  
Group Algebras ◽  
Prime Characteristic ◽  
Characteristic P

We prove that if the p-reduced abelian group G is a special countable extension of its totally projective p-component of torsion Gp and R is a perfect commutative unitary ring of prime characteristic p, then the group S(G) of all normed p-units in the group algebra RG modulo Gp, that is, S(G)/Gp, is totally projective. Our result strengthens both classical results obtained by May and Hill–Ullery.

scholarly journals Unit groups of group algebras of Abelian groups of order 32

2021 ◽  
Vol 40 (5) ◽  
pp. 1341-1356
Author(s):  
Suchi Bhatt ◽  
Harish Chandra
Keyword(s):  
Finite Field ◽  
Abelian Groups ◽  
Complete Characterization ◽  
Group Algebras ◽  
Characteristic P

Let F be a finite field of characteristic p > 0 with q = pn elements. In this paper, a complete characterization of the unit groups U(F G) of group algebras F G for the abelian groups of order 32, over finite field of characteristic p > 0 has been obtained.

Irreducible Modules for Polycyclic Group Algebras

1981 ◽  
Vol 33 (4) ◽  
pp. 901-914 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Nilpotent Group ◽  
Irreducible Module ◽  
Polycyclic Group ◽  
Group Algebras ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.We introduce some terminology.

Sign in / Sign up

Export Citation Format

Share Document