Group algebras of Lie nilpotency index 14

2019 ◽  
Vol 13 (05) ◽  
pp. 2050088
Author(s):  
Suchi Bhatt ◽  
Harish Chandra ◽  
Meena Sahai
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Nilpotency Index

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. Lie nilpotent group algebras of strong Lie nilpotency index at most 13 have been classified by many authors. In this paper, our aim is to classify the group algebras [Formula: see text] which are strongly Lie nilpotent of index 14.

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.

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.

Group algebras of Lie nilpotency index 12 and 13

2017 ◽  
Vol 46 (4) ◽  
pp. 1428-1446 ◽  
Author(s):  
R. K. Sharma ◽  
Reetu Siwach ◽  
Meena Sahai
Keyword(s):  
Group Algebras ◽  
Nilpotency Index

Modular Group Algebras with Small Upper Lie Nilpotency Index

2020 ◽  
Vol 12 (1) ◽  
pp. 108-111
Author(s):  
Suchi Bhatt ◽  
Harish Chandra
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Group Algebras ◽  
Modular Group Algebra ◽  
Nilpotency Index

Let KG be the modular group algebra of a group G over a field K of characteristic p > 0. The classification of group algebras KG with upper Lie nilpotency index tL(KG) greater than or equal to |G′| – 13p + 14 have already been done. In this paper, our aim is to classify the group algebras KG for which tL(KG) = |G′| – 14p + 15.

The nilpotency index of the radicals of group algebras of finite groups whose Sylow 3-subgroups are extra-special of order 27 of exponent 3

1988 ◽  
Vol 108 (1-2) ◽  
pp. 117-132
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Simple Group ◽  
Finite Groups ◽  
Jacobson Radical ◽  
Group Algebras ◽  
Nilpotency Index ◽  
Characteristic 3 ◽  
A Finite Group

SynopsisLet J(FG) be the Jacobson radical of the group algebra FG of a finite groupG with a Sylow 3-subgroup which is extra-special of order 27 of exponent 3 over a field F of characteristic 3, and let t(G) be the least positive integer t with J(FG)t = 0. In this paper, we prove that t(G) = 9 if G has a normal subgroup H such that (|G:H|, 3) = 1 and if H is either 3-solvable, SL(3,3) or the Tits simple group 2F4(2)'.

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.

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.

Lie Nilpotent Group Algebras and Upper Lie Codimension Subgroups

2006 ◽  
Vol 34 (10) ◽  
pp. 3859-3873 ◽  
Author(s):  
Francesco Catino ◽  
Ernesto Spinelli
Keyword(s):  
Nilpotent Group ◽  
Group Algebras

scholarly journals Free Symmetric and Unitary Pairs in the Field of Fractions of Torsion-Free Nilpotent Group Algebras

2019 ◽  
Vol 23 (3) ◽  
pp. 605-619
Author(s):  
Vitor O. Ferreira ◽  
Jairo Z. Gonçalves ◽  
Javier Sánchez
Keyword(s):  
Nilpotent Group ◽  
Group Algebras ◽  
Free Nilpotent Group ◽  
Torsion Free
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