Peyman Niroomand | Damghan University (original) (raw)

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Papers by Peyman Niroomand

Denoting by H⊗KH⊗K the nonabelian tensor product of two subgroups HH and KK of a finite group GG,... more Denoting by H⊗KH⊗K the nonabelian tensor product of two subgroups HH and KK of a finite group GG, we investigate the relative tensor degree d⊗(H,K)=|{(h,k)∈H×K|h⊗k=1}||H||K| of HH and KK. The case H=K=GH=K=G has been studied recently. Here we deal with arbitrary subgroups HH and KK, showing analogies and differences between d⊗(H,K)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 Erdős.

We prove a theorem of splitting for the nonabelian tensor product LotimesNL \otimes NLotimesN of a pair (L,N)(L,N)(L,N)... more We prove a theorem of splitting for the nonabelian tensor product LotimesNL \otimes NLotimesN of a pair (L,N)(L,N)(L,N) of Lie algebras LLL and NNN in terms of its diagonal ideal LsquareNL \square NLsquareN and of the nonabelian exterior product LwedgeNL \wedge NLwedgeN. A similar circumstance was described two years ago by the second author in the special case N=LN=LN=L. The interest is due to the fact that the size of LsquareNL \square NLsquareN influences strongly the structure of LotimesNL \otimes NLotimesN. Another question, often related to the structure of LotimesNL \otimes NLotimesN, deals with the behaviour of the operator square\squaresquare with respect to the formation of free products. We answer with another theorem of splitting even in this case, noting some connections with the homotopy theory.

To appear in Mathematical Reports

To appear in Turk. J. Math/DOI: 10.3906/mat-1302-50

Communications in Algebra, 2010

We introduce the exterior degree of a finite group G to be the probability for two elements g and... more 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.

In virtue of a recent bound obtained in [P. Niroomand, F.G. Russo, A note on the Schur multiplier... more 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).

"By a well-known result of Green \cite{gr} and the formal definition of \cite{el1}, there is an i... more "By a well-known result of Green \cite{gr} and the formal definition of \cite{el1}, there is an integer ttt, say corank$(G)$, such that ∣mathcalM(G)∣=pf12n(n−1)−t|\mathcal{M}(G)|=p^{\f12n(n-1)-t}∣mathcalM(G)∣=pf12n(n−1)−t.
In \cite{ni}, the author showed for a non-abelian group GGG, corank$(G)\geq\log_p(|G|)-2$ and classified the structure of all non-abelian ppp-groups of corank logp(∣G∣)−2\log_p(|G|)-2logp(∣G∣)−2. In the present paper, we are interesting to characterize the structure of all ppp-groups of corank logp(∣G∣)−1\log_p(|G|)-1logp(∣G∣)−1."

Journal of Algebra and Its Applications, Nov 2012

Arxiv preprint arXiv:1002.2563, Jan 1, 2010

For every finite p-group G of order p n with derived subgroup of order p m , Rocco [N.R. Rocco, O... more For every finite p-group G of order p n with derived subgroup of order p m , 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 p n(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.

The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of group... more The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of groups of square free order and of groups of orders p2qp^2qp2q and p2qrp^2qrp2qr, where ppp, qqq and rrr are primes and p<q<rp<q<rp<q<r.

Denoting by H⊗KH⊗K the nonabelian tensor product of two subgroups HH and KK of a finite group GG,... more Denoting by H⊗KH⊗K the nonabelian tensor product of two subgroups HH and KK of a finite group GG, we investigate the relative tensor degree d⊗(H,K)=|{(h,k)∈H×K|h⊗k=1}||H||K| of HH and KK. The case H=K=GH=K=G has been studied recently. Here we deal with arbitrary subgroups HH and KK, showing analogies and differences between d⊗(H,K)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 Erdős.

We prove a theorem of splitting for the nonabelian tensor product LotimesNL \otimes NLotimesN of a pair (L,N)(L,N)(L,N)... more We prove a theorem of splitting for the nonabelian tensor product LotimesNL \otimes NLotimesN of a pair (L,N)(L,N)(L,N) of Lie algebras LLL and NNN in terms of its diagonal ideal LsquareNL \square NLsquareN and of the nonabelian exterior product LwedgeNL \wedge NLwedgeN. A similar circumstance was described two years ago by the second author in the special case N=LN=LN=L. The interest is due to the fact that the size of LsquareNL \square NLsquareN influences strongly the structure of LotimesNL \otimes NLotimesN. Another question, often related to the structure of LotimesNL \otimes NLotimesN, deals with the behaviour of the operator square\squaresquare with respect to the formation of free products. We answer with another theorem of splitting even in this case, noting some connections with the homotopy theory.

To appear in Mathematical Reports

To appear in Turk. J. Math/DOI: 10.3906/mat-1302-50

Communications in Algebra, 2010

We introduce the exterior degree of a finite group G to be the probability for two elements g and... more 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.

In virtue of a recent bound obtained in [P. Niroomand, F.G. Russo, A note on the Schur multiplier... more 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).

"By a well-known result of Green \cite{gr} and the formal definition of \cite{el1}, there is an i... more "By a well-known result of Green \cite{gr} and the formal definition of \cite{el1}, there is an integer ttt, say corank$(G)$, such that ∣mathcalM(G)∣=pf12n(n−1)−t|\mathcal{M}(G)|=p^{\f12n(n-1)-t}∣mathcalM(G)∣=pf12n(n−1)−t.
In \cite{ni}, the author showed for a non-abelian group GGG, corank$(G)\geq\log_p(|G|)-2$ and classified the structure of all non-abelian ppp-groups of corank logp(∣G∣)−2\log_p(|G|)-2logp(∣G∣)−2. In the present paper, we are interesting to characterize the structure of all ppp-groups of corank logp(∣G∣)−1\log_p(|G|)-1logp(∣G∣)−1."

Journal of Algebra and Its Applications, Nov 2012

Arxiv preprint arXiv:1002.2563, Jan 1, 2010

For every finite p-group G of order p n with derived subgroup of order p m , Rocco [N.R. Rocco, O... more For every finite p-group G of order p n with derived subgroup of order p m , 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 p n(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.

The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of group... more The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of groups of square free order and of groups of orders p2qp^2qp2q and p2qrp^2qrp2qr, where ppp, qqq and rrr are primes and p<q<rp<q<rp<q<r.

C. R. Math. Acad. Sci. Paris, Ser. I

It has been proved in J.A. Green (1956) [5] for every p-group of order pn, |M(G)| =p^1/2 n(n−1)−t... more It has been proved in J.A. Green (1956) [5] for every p-group of order pn, |M(G)| =p^1/2 n(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].