Peyman Niroomand

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.

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.

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

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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".

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.