Modular Group Algebras Research Articles

On the modular isomorphism problem for 2-generated groups with cyclic derived subgroup

We continue the analysis of the modular isomorphism problem for [Formula: see text]-generated [Formula: see text]-groups with cyclic derived subgroup, [Formula: see text], started in [D. García-Lucas, Á. del Río and M. Stanojkowski, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebra Represent. Theory 26 (2022) 2683–2707, doi:10.1007/s10468-022-10182-x]. We show that if [Formula: see text] belongs to this class of groups, then the isomorphism type of the quotients [Formula: see text] and [Formula: see text] are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification [O. Broche, D. García-Lucas and Á. del Río, A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order, Int. J. Algebra Comput. 33(4) (2023) 641–686]. We also show that for groups in this class of order at most [Formula: see text], the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order [Formula: see text] whose group algebras over the field with [Formula: see text] elements cannot be distinguished with the techniques available to us.

Abelian invariants and a reduction theorem for the modular isomorphism problem

We show that elementary abelian direct factors can be disregarded in the study of the modular isomorphism problem. Moreover, we obtain four new series of abelian invariants of the group base in the modular group algebra of a finite p-group. Finally, we apply our results to new classes of groups.

A reduction theorem for the Isomorphism Problem of group algebras over fields

We prove that the Isomorphism Problem for group algebras reduces to group algebras over finite extensions of the prime field. In particular, the Modular Isomorphism Problem reduces to finite modular group algebras.

Modular group algebras with centrally metabelian unit groups

Let F be a field of odd characteristic and G a group such that the group algebra FG is modular. We obtain necessary and sufficient conditions for the unit group of FG to be centrally metabelian.

Unit groups of commutative modular group algebras

This paper analyzes some of the fundamental results of the unit groups of the commutative modular group algebras. We introduce a more visible and concise form of a number of these results. We consider with a corresponding justification some essentially inexact and unproved results in some papers and for certain of them either we mark the corrections or we point out the corrections, which are founded in other articles.

On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document}. We prove that the exponent and the abelianization of the centralizer of G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document} in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document} is determined. These claims are known to be false for p = 2.

Modular group algebras whose group of unitary units is locally nilpotent

We characterize those modular group algebras FG whose group of unitary units is locally nilpotent or nilpotent under the classical involution of FG. The main results of [19, 20] are trivial consequences of this note.

DECODING BINARY REED-MULLER CODES VIA GROEBNER BASES

The binary Reed-Muller codes can be characterized as the radical powers of a modular group algebra. In this paper, we deal with the Groebner bases to decode these codes.

On algebraic structure of the Reed-Muller codes

It is known that the Reed-Muller codes over a prime field may be described as the radical powers of a modular group algebra. In this paper, we give a new proof of the same result in a quotient of a polynomial ring. Special elements in a prime field are studied. An interpolation polynomial is introduced in order to characterize the coefficients of the Jennings polynomials.

On the normal complement problem in modular and semisimple group algebras

Let p and q be odd primes such that Let F be the field with p elements and be a group, where A is an abelian group of order In this article, we prove that if then G does not have a normal complement in Further, for any integer we prove that if F is a finite field such that then and do not have a normal complement in and respectively.

A note on modular group algebras with upper Lie nilpotency indices

Let KG be the modular group algebra of anarbitrary group G over a field K of characteristic p>0. In thispaper we give some improvements of upper Lie nilpotency indext L(KG) of the group algebra KG. It can be seen that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is atleast p+1. In this way the classification of group algebras KG with next upper Lie nilpotency indext L(KG) up to 9p−7 have alreadybeen classified. Furthermore, we give a complete classification ofmodular group algebraKGfor which the upper Lie nilpotency index is 10p−8.

Non-isomorphic 2-groups with isomorphic modular group algebras

Abstract We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.

Centre of Unitary Subgroup of Modular Group Algebra’s

We establish the structure of the centre of V∗(F2 k (M2n+1 )), 1 V∗(F2 k (M2n+1 × C2)) and V∗(F2 k ((M2n+1 × C2) × C2)) over a finite field of characteristics 2 where M2n+1 =< ψ, λ|ψ 2n = λ 2 = 1, λψ = ψ 2n+1λ > is the Modular group having order 2 n+1 and C2 is a cyclic group of order 2

On the modular isomorphism problem for groups of class 3 and obelisks

Abstract We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.

Lie nilpotency index of a modular group algebra

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text] for [Formula: see text] and [Formula: see text]. Group algebras of upper Lie nilpotency index [Formula: see text] for [Formula: see text], have already been characterized completely.

The isomorphism problem of unitary subgroups of modular group algebras

The isomorphism problem of unitary subgroups of modular group algebras

Modular Group Algebras with Small Upper Lie 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.

A note on the upper Lie nilpotency index of a group algebra

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text]. The group algebra [Formula: see text] with [Formula: see text] has already been described.

A note on normal complement problem for split metacyclic groups

In this article, we discuss the normal complement problem for metacyclic groups in modular group algebras. If F is the field with p elements and G is a finite split metacyclic p-group of nilpotency class 2, then we prove that G has a normal complement in For a finite field F of characteristic p, where p is an odd prime, we prove that has a normal complement in if and only if p = 3 and

Lie nilpotency indices of modular group algebras II

Let [Formula: see text] be the modular group algebra of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text]. The classification of the group algebras [Formula: see text] with upper Lie nilpotency index [Formula: see text] greater than or equal to [Formula: see text] has already been done. In this paper, we determine the group algebras [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].