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Peyman Niroomand

Associate Professor of Pure Mathematics

Education

  • Ph.D. 2004-2008

    Relative covering groups and some properties on tensor centre of group

    Ferdowsi University of Mashhad. Mashhad, Iran

  • M.Sc. 2002-2004

    Relative covering groups

    Ferdowsi University of Mashhad. Mashhad, Iran

  • B.Sc.1998-2002

    Pure Mathematics

    Shahid Bahonar University of Kerman, Kerman, Iran

Teaching

  • Algebra I,II,III
  • Finite p-groups and their applications
  • Schur multiplier
  • Lie algebras

Selected Publications

Ghorbanzadeh, T.J., Parvizi, M., Niroomand, P. The non-Abelian tensor square of p-groups of order p4 (2018) Asian-European Journal of Mathematics, 11 (6), art. no. 1850084, .

DOI: 10.1142/S1793557118500845

In this paper, in the class of p-groups of order p4, we obtain the non-Abelian exterior square, the exterior center, the non-Abelian tensor square, the tensor center and the third homotopy group of suspension of an Eilenberg-MacLane space k(G, 1) of such groups. © 2018 World Scientific Publishing Company.

AUTHOR KEYWORDS: Non-Abelian tensor square; p-group
PUBLISHER: World Scientific Publishing Co. Pte Ltd

Niroomand, P., Johari, F. The structure, capability and the Schur multiplier of generalized Heisenberg Lie algebras (2018) Journal of Algebra, 505, pp. 482-489.

DOI: 10.1016/j.jalgebra.2018.03.014

From Berkovich and Janko [3, Problem 1729] asked to obtain the Schur multiplier and the representation of a group G, when G is a special p-group minimally generated by d elements and |G′|=p[Formula presented]d(d−1). Here, we intend to give an answer to this question similarly for nilpotent Lie algebras. Furthermore, we give some results about the tensor square and the Schur multiplier of some nilpotent Lie algebras of class two. © 2018 Elsevier Inc.

AUTHOR KEYWORDS: Capability; Generalized Heisenberg; Nilpotent Lie algebra; Schur multiplier
PUBLISHER: Academic Press Inc.

Niroomand, P., Johari, F., Parvizi, M. Capable Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field (2018) Linear and Multilinear Algebra, pp. 1-13. Article in Press.

DOI: 10.1080/03081087.2018.1425356

In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given. © 2018 Informa UK Limited, trading as Taylor & Francis Group

AUTHOR KEYWORDS: Capability; generalized Heisenberg Lie algebras; Schur multiplier; stem Lie algebras
PUBLISHER: Taylor and Francis Ltd.

Niroomand, P. Classifying P-groups by their schur multipliers (2018) Mathematical Reports, 20 (3), pp. 279-284.

Some recent results devoted to the investigation of the structure of p-groups rely on the study of their Schur multipliers. One of these results states that for any p-group G of order pn there exists a nonnegative integer s(G) such that the order of the Schur multiplier of G is equal to p12 (n−1)(n−2)+1−s(G). Characterizations of the structure of all non-abelian p-groups G have been obtained for the case that s(G) = 0 or 1. The present paper is devoted to the characterization of all p-groups with s(G) = 2. © 2018 Editura Academiei Romane. All rights reserved.

AUTHOR KEYWORDS: P-groups; Schur multiplier
PUBLISHER: Editura Academiei Romane

Divandari, M., Shanbehbazari, F.P., Niroomand, P., Faramarzi Salles, A. On finite groups with a given number of exterior centralizers (2018) Communications in Algebra, . Article in Press.

DOI: 10.1080/00927872.2018.1469030

A group G is called n-centralizer if it has n distinct centralizers. In this paper, in analogs to n-centralizer, we say a group G is n-exterior centralizer provided G has n distinct exterior centralizers. The current paper is devoted to characterize all groups that are n-exterior centralizer, where n ε {1,2,3,4,5}. © 2018, © 2018 Taylor & Francis Group, LLC.

AUTHOR KEYWORDS: Centralizer; exterior centralizer; non-Abelian exterior products; Schur multiplier
PUBLISHER: Taylor and Francis Inc.

Saeedi, F., Arabyani, H., Niroomand, P. On dimension of Schur multiplier of nilpotent Lie algebras II (2017) Asian-European Journal of Mathematics, 10 (4), art. no. 1750076, .

DOI: 10.1142/S1793557117500760

Let L be a non-abelian nilpotent Lie algebra of dimension n and put s(L) = 1/2(n - 1)(n - 2) + 1-dim M(L), where M(L) denotes the Schur multiplier of L. Niroomand and Russo in 2011 proved that s(L) ≥ 0 and that s(L) = 0 if and only if L ≅ H(1) ⊕ F(n - 3), in which H(1) is the Heisenberg algebra of dimension 3 and F(n - 3) is the abelian (n - 3)-dimensional Lie algebra. In the same year, they also classified all nilpotent Lie algebras L satisfying s(L) = 1 or 2. In this paper, we obtain all nilpotent Lie algebras L provided that s(L) = 3. © 2017 World Scientific Publishing Company.

AUTHOR KEYWORDS: Multiplier; nilpotent Lie algebra
PUBLISHER: World Scientific Publishing Co. Pte Ltd

Niroomand, P., Parvizi, M. 2-capability and 2-nilpotent multiplier of finite dimensional nilpotent Lie algebras (2017) Journal of Geometry and Physics, 121, pp. 180-185.

DOI: 10.1016/j.geomphys.2017.07.003

In the present context, we investigate to obtain some more results about 2-nilpotent multiplier M(2)(L) of a finite dimensional nilpotent Lie algebra L. For instance, we characterize the structure of M(2)(H) when H is a Heisenberg Lie algebra. Moreover, we give some inequalities on dimM(2)(L) to reduce a well known upper bound on 2-nilpotent multiplier as much as possible. Finally, we show that H(m) is 2-capable if and only if m=1. © 2017 Elsevier B.V.

AUTHOR KEYWORDS: 2-capable Lie algebra; 2-nilpotent multiplier; Derived subalgebra; Heisenberg algebras
PUBLISHER: Elsevier B.V.

Niroomand, P., Russo, F.G. Corrigendum to “Probabilistic properties of the relative tensor degree of finite groups” (Indagationes Mathematicae (2016) 27(1) (147–159) (S0019357715000695) (10.1016/j.indag.2015.09.002)) (2017) Indagationes Mathematicae, 28 (2), pp. 612-614.

DOI: 10.1016/j.indag.2016.08.003

We clarify Theorems 5.1 and 5.4 of our paper “Probabilistic properties of the relative tensor degree of finite groups”, published in Indagationes Mathematicae 27 (2016), 147–159. © 2016

AUTHOR KEYWORDS: Commutativity degree; Exterior degree; Relative tensor degree
PUBLISHER: Elsevier B.V.

Johari, F., Parvizi, M., Niroomand, P. Capability and Schur multiplier of a pair of Lie algebras (2017) Journal of Geometry and Physics, 114, pp. 184-196.

DOI: 10.1016/j.geomphys.2016.11.016

The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We characterize the exact structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg ones. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras. © 2016 Elsevier B.V.

AUTHOR KEYWORDS: Capable Lie algebra; Capable pair of Lie algebras; Exterior product; Schur multiplier; Tensor product
PUBLISHER: Elsevier B.V.

Niroomand, P., Russo, F.G. On the tensor degree of finite groups (2017) Ars Combinatoria, 131, pp. 273-283.

We study the number of elements x and y of a finite group G such that x⊗ y = 1G⊗G in the nonabelian tensor square G⊗G of G. This number, divided by |G|2, is called the tensor degree of G and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335-343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.

AUTHOR KEYWORDS: Commutativity degree; Dihedral groups; Exterior degree; P-groups; Schur multiplier; Tensor degree
PUBLISHER: Charles Babbage Research Centre

Niroomand, P., Erfanian, A., Parvizi, M., Tolue, B. Non-exterior square graph of finite group (2017) Filomat, 31 (3), pp. 877-883.

DOI: 10.2298/FIL1703877N

We define the non-exterior square graph ˆΓG which is a graph associated to a non-cyclic finite group with the vertex set GẐ(G), where Ẑ(G) denotes the exterior centre of G, and two vertices x and y are joined whenever x ∧ y ≠ 1, where ∧ denotes the operator of non-abelian exterior square. In this paper, we investigate how the group structure can be affected by the planarity, completeness and regularity of this graph. © 2017, University of Nis. All rights reserved.

AUTHOR KEYWORDS: Capable groups; Exterior degree; Non-commutating graph; Non-exterior square graph; Schur multiplier
PUBLISHER: University of Nis

Niroomand, P., Parvizi, M. 2-Nilpotent multipliers of a direct product of Lie algebras (2016) Rendiconti del Circolo Matematico di Palermo, 65 (3), pp. 519-523.

DOI: 10.1007/s12215-016-0251-0

In this paper, we present an explicit formula for the 2-nilpotent multiplier of a direct product of two Lie algebras. © 2016, Springer-Verlag Italia.

PUBLISHER: Springer-Verlag Italia s.r.l.

Niroomand, P., Johari, F., Parvizi, M. On the capability and schur multiplier of nilpotent lie algebra of class two (2016) Proceedings of the American Mathematical Society, 144 (10), pp. 4157-4168.

DOI: 10.1090/proc/13092

Recently, the authors in a joint paper obtained the structure of all capable nilpotent Lie algebras with derived subalgebra of dimension at most 1. This paper is devoted to characterizing all capable nilpotent Lie algebras of class two with derived subalgebra of dimension 2. It develops and generalizes the result due to Heineken for the group case. © 2016 American Mathematical Society.

AUTHOR KEYWORDS: Capable lie algebra; Nilpotent lie algebra; Schur multiplier
PUBLISHER: American Mathematical Society

Niroomand, P. Some results on the exterior degree of extra-special groups (2016) Ars Combinatoria, 125, pp. 121-128.

The concept of exterior degree of a finite group C is introduced by the author in a joint paper [13] which is the probability of randomly two elements g and h in G such that gδ = 1. In the present paper, a necessary and sufficient condition for a non cyclic group is given when its exterior degree achieves the upper bound (p2 + p - 1)/p3 in which p is the smallest prime number dividing the order of C. We also compute the exterior degree of all extra-special p-groups. Finally, for an extra-special p-group H and a group C when G/Zδ (G) is p-group, we will show that dδ (C) = dδ (H) if and only if G/ZA(G)∼H/Zδ (H) provided that dA(C) 11/32.

AUTHOR KEYWORDS: Capable group; Schur multiplier; extra-special group.; Commutativity degree; Exterior centre; Exterior degree; Phrases
PUBLISHER: Charles Babbage Research Centre

Niroomand, P., Russo, F.G. Probabilistic properties of the relative tensor degree of finite groups (2016) Indagationes Mathematicae, 27 (1), pp. 147-159.

DOI: 10.1016/j.indag.2015.09.002

Denoting by H ⊗ K the nonabelian tensor product of two subgroups H and K of a finite group G, we investigate the relative tensor degree d⊗(H,K)=|{(h,k)∈H × K|h⊗k=1}||H||K| of H and K. The case H=K=G has been studied recently. Here we deal with arbitrary subgroups H and K, showing analogies and differences between d⊗(H, K) and the relative commutativity degree d(H,K)=|{(h,k)∈H×K|[h,k]=1}||H||K|, which is a generalization of the probability of commuting elements, introduced by Erdos. © 2015 Royal Dutch Mathematical Society (KWG).

AUTHOR KEYWORDS: Commutativity degree; Exterior degree; Relative tensor degree
PUBLISHER: Elsevier

Niroomand, P. Characterizing finite p-groups by their Schur multipliers, t(G) = 5 (2015) Mathematical Reports, 17 (2), pp. 249-254.

Let G be a finite p-group of order pn. It is known that |M(G)| = p1/2n(n-1)-t(G) and t(G) ≥ 0. The structure of G for t(G) ≥ 4 was determined by several authors. In this paper we will describe all the possible structures of G for t(G) = 5.

AUTHOR KEYWORDS: P-groups; Schur multiplier
PUBLISHER: Editura Academiei Romane

Niroomand, P., Parvizi, M. On the 2-nilpotent multiplier of finite p-groups (2015) Glasgow Mathematical Journal, 57 (1), pp. 201-210.

DOI: 10.1017/S0017089514000263

The purpose of this paper is a further investigation on the 2-nilpotent multiplier, M (2)(G), when G is a non-abelian p-group. Furthermore, taking G in the class of extra-special p-groups, we will get the explicit structure of Ms (2)(G) and will classify 2-capable groups in that class. © 2014 Glasgow Mathematical Journal Trust.

PUBLISHER: Cambridge University Press

Niroomand, P., Russo, F.G. On the size of the third homotopy group of the suspension of an Eilenberg-MacLane space (2014) Turkish Journal of Mathematics, 38 (4), pp. 664-671.

DOI: 10.3906/mat-1302-50

The nonabelian tensor square G ⊗ G of a group G of |G| = pn and |G'| = pm (p prime and n,m ≥ 1) satisfies a classic bound of the form |G ⊗ G| ≤ pn(n-m). This allows us to give an upper bound for the order of the third homotopy group π3(SK(G,1)) of the suspension of an Eilenberg-MacLane space K(G,1), because π3(K(G,1)) is isomorphic to the kernel of k:x ⊗ y ∈ G ⊗ G {mapping} [x,y] ∈ G'. We prove that |G ⊗ G| ≤ p(n-1)(n-m)+2, sharpening not only |G ⊗ G| ≤ pn(n-m) but also supporting a recent result of Jafari on the topic. Consequently, we discuss restrictions on the size of π3(SK(G,1)) based on this new estimation. © Tübitak.

AUTHOR KEYWORDS: Homotopy; Nonabelian tensor square; P-groups; Schur multipliers
PUBLISHER: TUBITAK

Rezaei, R., Niroomand, P., Erfanian, A. On the multiple exterior degree of finite groups (2014) Mathematica Slovaca, 64 (4), pp. 859-870.

DOI: 10.2478/s12175-014-0244-4

Recently the first two authors have introduced a group invariant, called exterior degree, which is related to the number of elements x and y of a finite group G such that xΛy = 1 in the exterior square GΛG of G. Research on this topic gives some relations between this concept, the Schur multiplier and the capability of a finite group. In the present paper, we will generalize the concept of exterior degree of groups and we will introduce the multiple exterior degree of finite groups. Among other results, we will obtain some relations between the multiple exterior degree, multiple commutativity degree and capability of finite groups. © 2014, Versita Warsaw and Springer-Verlag Wien.

AUTHOR KEYWORDS: capable groups; exterior degree; multiple commutativity degree; multiple exterior degree; Schur multiplier
PUBLISHER: Versita

Niroomand, P., Parvizi, M., Russo, F.G. Some criteria for detecting capable Lie algebras (2013) Journal of Algebra, 384, pp. 36-44.

DOI: 10.1016/j.jalgebra.2013.02.033

In virtue of a recent bound obtained in [P. Niroomand, F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011) 1293-1297], we classify all capable nilpotent Lie algebras of finite dimension possessing a derived subalgebra of dimension one. Indirectly, we find also a criterion for detecting noncapable Lie algebras. The final part contains a construction, which shows that there exist capable Lie algebras of arbitrary big corank (in the sense of Berkovich-Zhou). © 2013 Elsevier Inc.

AUTHOR KEYWORDS: Capable Lie algebras; Corank; Exterior product; Schur multipliers

Niroomand, P., Rezaei, R. The Exterior Degree of a Pair of Finite Groups (2013) Mediterranean Journal of Mathematics, 10 (3), pp. 1195-1206.

DOI: 10.1007/s00009-013-0252-6

The exterior degree of a pair of finite groups (G, N), which is a generalization of the exterior degree of finite groups, is the probability for two elements (g, n) in (G, N) such that g ∧ n = 1. In the present paper, we state some relations between this concept and the relative commutatively degree, capability and the Schur multiplier of a pair of groups. © 2013 Springer Basel.

AUTHOR KEYWORDS: capability of groups; commutativity degree; Exterior degree; pair of groups; Schur multiplier

Niroomand, P., Russo, F.G. An improvement of a bound of green (2012) Journal of Algebra and its Applications, 11 (6), art. no. 12501162, .

DOI: 10.1142/S0219498812501162

A p-group G of order p n (p prime, n < 1) satisfies a classic Green's bound log p |M(G)| ≤ 12;n(n - 1) on the order of the Schur multiplier M(G) of G. Ellis and Wiegold sharpened this restriction, proving that log p |M(G)| ≤ 12;(d - 1)(n + m), where |G′| = p m (m < 1) and d is the minimal number of generators of G. The first author has recently shown that log p |M(G)| ≤ 12;(n + m - 2)(n - m - 1) + 1, improving not only Green's bound, but several other inequalities on |M(G)| in literature. Our main results deal with estimations with respect to the bound of Ellis and Wiegold. © 2012 World Scientific Publishing Company. [/accordion]
AUTHOR KEYWORDS: corank; p-groups; Schur multiplier

Jafari, S.H., Niroomand, P., Erfanian, A. The non-abelian tensor square and schur multiplier of groups of orders p 2q and p 2qr (2012) Algebra Colloquium, 19 (SPL. ISS. 1), pp. 1083-1088.

The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of groups of square-free orders and of groups of orders p 2q, pq 2 and p 2qr, where p, q and r are primes and p < q < r. © 2012 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University. [/accordion]
AUTHOR KEYWORDS: non-abelian tensor square; Schur multiplier

Niroomand, P. A note on the Schur multiplier of groups of prime power order (2012) Ricerche di Matematica, 61 (2), pp. 341-346.

DOI: 10.1007/s11587-012-0134-4

By a well-known result of Green (Proc R Soc A 237:574-581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191-196, 1999), there is an integer t, say corank(G), such that {pipe}M(G){pipe} = p 1/2n(n-1)-t. In Niroomand (J Algebra 322:4479-4482, 2009), the author showed for a non-abelian group G, corank(G) ≥ log p({pipe}G{pipe})-2 and classified the structure of all non-abelian p-groups of corank log p({pipe}G{pipe})-2. In the present paper, we are interesting to characterize the structure of all p-groups of corank log p({pipe}G{pipe})-1. © 2012 Università degli Studi di Napoli "Federico II".

AUTHOR KEYWORDS: p-group; Schur multiplier

Niroomand, P. Characterizing finite p-groups by their Schur multipliers (2012) Comptes Rendus Mathematique, 350 (19-20), pp. 867-870.

DOI: 10.1016/j.crma.2012.10.018

It has been proved in J.A. Green (1956) [5] for every p-group of order p n, |M(G)|=p12n(n-1)-t(G), where t(G)≥0. In Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], and X. Zhou (1994) [14], the structure of G has been characterized for t(G)=0, 1, 2, 3 by several authors. Also in A.R. Salemkar et al. (2007) [12], the structure of G characterized when t(G)=4 and Z(G) is elementary abelian, but there are some missing points in classifying the structure of these groups. This paper is devoted to classify the structure of G when t(G)=4 without any condition and with a short and quite different way to that of Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], A.R. Salemkar et al. (2007) [12], and X. Zhou (1994) [14]. © 2012 Académie des sciences.

Niroomand, P. Some properties on the tensor square of Lie algebras (2012) Journal of Algebra and its Applications, 11 (5), art. no. 1250085, .

DOI: 10.1142/S0219498812500855

In the present paper we extend the results of [2, 4] for the tensor square of Lie algebras. More precisely, for any Lie algebra L with L/L 2 of finite dimension, we prove L ⊗ L ≅ L □ L ⊕ L ∧ L and Z ∧(L) ∩ L 2 = Z ⊗(L). Moreover, we show that L ∧ L is isomorphic to derived subalgebra of a cover of L, and finally we give a free presentation for it. © 2012 World Scientific Publishing Company.

AUTHOR KEYWORDS: Schur multiplier of Lie algebra; Tensor square of Lie algebra

Niroomand, P., Rezaei, R., Russo, F.G. Commuting powers and exterior degree of finite groups (2012) Journal of the Korean Mathematical Society, 49 (4), pp. 855-865.

DOI: 10.4134/JKMS.2012.49.4.855

Recently, we have introduced a group invariant, which is related to the number of elements x and y of a finite group G such that x ∧ y = 1 G∧G in the exterior square G ∧ G of G. This number gives restrictions on the Schur multiplier of G and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form h m ∧ k of H ∧ K such that h m ∧ k = 1 H∧K, where m ≥ 1 and H and K are arbitrary subgroups of G. ©2012 The Korean Mathematical Society.

AUTHOR KEYWORDS: Commutativity degree; Exterior product; Homological algebra; m-th relative exterior degree; Schur multiplier

Moghaddam, M.R.R., Niroomand, P. Some Properties of Certain Subgroups of Tensor Squares of p-Groups (2012) Communications in Algebra, 40 (3), pp. 1188-1193.

DOI: 10.1080/00927872.2010.546185

Let G be a finite p-group of order p n and ∇(G) be the subgroup of the tensor square of G generated by all symbols x ⊗ x, for all x in G. In the present article, we construct an upper bound for the order of ∇(G) and any extra special p-group. It is also shown that ∇(G) ≅ ∇(G/G′). Using our result, we obtain the explicit structure of the tensor square of G and π 3SK(G, 1). Finally, the structure of G will be characterized when the bound is attained. © 2012 Copyright Taylor and Francis Group, LLC.

AUTHOR KEYWORDS: Extra special p-group; Non-abelian p-groups; Tensor square of groups

Parvizi, M., Niroomand, P. On the structure of groups whose exterior or tensor square is a p-group (2012) Journal of Algebra, 352 (1), pp. 347-353.

DOI: 10.1016/j.jalgebra.2011.12.001

It is well known that if G is a nilpotent (infinite) p-group of bounded exponent, then G⊗G (resp. G∧G) is also an (infinite) p-group. We study the converse under some restrictions. We also prove that if G is a finitely generated group with G ab capable, then the finiteness of G∧G implies that of G. © 2011 Elsevier Inc.

AUTHOR KEYWORDS: Capability; Exterior square; Locally finite groups; P-groups; Relative Schur multiplier; Schur multiplier; Tensor square

Niroomand, P. On the tensor square of non-Abelian nilpotent finite-dimensional lie algebras (2011) Linear and Multilinear Algebra, 59 (8), pp. 831-836.

DOI: 10.1080/03081087.2010.497491

For every finite p-group G of order pn with derived subgroup of order pm, Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63-79] proved that the order of tensor square of G is at most pn(n-m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L ⊗ L) ≤ (n-m)(n-1) + 2. Furthermore for m = 1, the explicit structure of L is given when the equality holds. © 2011 Taylor & Francis.

AUTHOR KEYWORDS: Nilpotent lie algebra; Schur multiplier; Tensor square

Niroomand, P., Russo, F.G. A note on the Schur multiplier of a nilpotent Lie algebra (2011) Communications in Algebra, 39 (4), pp. 1293-1297.

DOI: 10.1080/00927871003652660

For a nilpotent Lie algebra L of dimension n and dim(L2) = m ≥ 1, we find the upper bound dim(M(L)) ≤ 1/2 (n + m - 2)(n - m - 1) + 1, where M(L) denotes the Schur multiplier of L. In case m = 1, the equality holds if and only if L ≅ H(1) ⊕ A, where A is an abelian Lie algebra of dimension n - 3 and H(1) is the Heisenberg algebra of dimension 3. © Taylor & Francis Group, LLC.

AUTHOR KEYWORDS: Nilpotent lie algebras; Schur multiplier

Niroomand, P., Russo, F.G. A restriction on the schur multiplier of nilpotent lie algebras (2011) Electronic Journal of Linear Algebra, 22, pp. 1-9.

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L ⊗ L are illustrated.

AUTHOR KEYWORDS: Derived subalgebra; Nilpotent lie algebras; Non-abelian tensor product; Schur multiplier

Niroomand, P. On dimension of the Schur multiplier of nilpotent Lie algebras (2011) Central European Journal of Mathematics, 9 (1), pp. 57-64.

DOI: 10.2478/s11533-010-0079-3

Let L be an n-dimensional non-abelian nilpotent Lie algebra and s(L) = (n - 1)(n - 2) + 1 - dim M(L) where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2. © 2011 Versita Warsaw and Springer-Verlag Wien.

AUTHOR KEYWORDS: Nilpotent Lie algebras; Schur multiplier

Niroomand, P., Rezaei, R. On the exterior degree of finite groups (2011) Communications in Algebra, 39 (1), pp. 335-343.

DOI: 10.1080/00927870903527568

We introduce the exterior degree of a finite group G to be the probability for two elements g and g′ in G such that g ∧ g′ = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier. Copyright © Taylor & Francis Group, LLC.

AUTHOR KEYWORDS: Capable group; Commutativity degree; Exterior centre; Exterior degree; Schur multiplier

Niroomand, P. The Schur multiplier of p-groups with large derived subgroup (2010) Archiv der Mathematik, 95 (2), pp. 101-103.

DOI: 10.1007/s00013-010-0154-9

For any given p-group of order pn (n ≥ 4) with derived subgroup of order pn-2 we will show that the order of its Schur multiplier is less than {pipe}G'{pipe}/2 when p = 2 and {pipe}G'{pipe} in the other cases. © 2010 Springer Basel AG.

AUTHOR KEYWORDS: p-groups; Schur multiplier

Niroomand, P. The converse of Schur's theorem (2010) Archiv der Mathematik, 94 (5), pp. 401-403.

DOI: 10.1007/s00013-010-0106-4

Schur's theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have {pipe}G/Z(G){pipe} ≤ {pipe}G′{pipe}d(G/Z(G)). In the special case of G being nilpotent, we prove {pipe}G/Z(G){pipe} divides {pipe}G′{pipe}d(G/Z(G)). © 2010 Birkhäuser/Springer Basel AG.

AUTHOR KEYWORDS: Nilpotent group; Schur's theorem

Niroomand, P. On the order of Schur multiplier of non-abelian p-groups (2009) Journal of Algebra, 322 (12), pp. 4479-4482.

DOI: 10.1016/j.jalgebra.2009.09.030

Let G be a finite p-group of order pn, Green proved that M (G), its Schur multiplier is of order at most pfrac(1, 2) n (n - 1). Later Berkovich showed that the equality holds if and only if G is elementary abelian of order pn. In the present paper, we prove that if G is a non-abelian p-group of order pn with derived subgroup of order pk, then | M (G) | ≤ pfrac(1, 2) (n + k - 2) (n - k - 1) + 1. In particular, | M (G) | ≤ pfrac(1, 2) (n - 1) (n - 2) + 1, and the equality holds in this last bound if and only if G = H × Z, where H is extra special of order p3 and exponent p, and Z is an elementary abelian p-group. © 2009 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Non-abelian p-groups; Schur multiplier

Niroomand, P., Russo, F. A note on the exterior centralizer (2009) Archiv der Mathematik, 93 (6), pp. 505-512.

DOI: 10.1007/s00013-009-0077-5

The notion of the exterior centralizer CG(x) of an element x of a group G is introduced in the present paper in order to improve some known results on the non-abelian tensor product of two groups. We study the structure of G by looking at that of CG(x) and we find some bounds for the Schur multiplier M(G) of G. © 2009 Birkhäuser Verlag Basel/Switzerland.

AUTHOR KEYWORDS: Exterior center; Exterior centralizer; Non-abelian tensor product; Schur multiplier

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