scholarly journals On the non-abelian tensor square of all groups of order dividing p5

2020 ◽  
pp. 2150107
Author(s):  
Taleea Jalaeeyan Ghorbanzadeh ◽  
Mohsen Parvizi ◽  
Peyman Niroomand
Keyword(s):  
Homotopy Group ◽  
The Third ◽  
Tensor Square ◽  
Exterior Square

In this paper, we consider all groups of order dividing [Formula: see text]. We obtain the explicit structure of the non-abelian tensor square, non-abelian exterior square, tensor center, exterior center, the third homotopy group of suspension of an Eilenberg–MacLane space [Formula: see text] and [Formula: see text] of such groups.

The non-abelian tensor square of p-groups of order p4

2018 ◽  
Vol 11 (06) ◽  
pp. 1850084
Author(s):  
Taleea Jalaeeyan Ghorbanzadeh ◽  
Mohsen Parvizi ◽  
Peyman Niroomand
Keyword(s):  
Homotopy Group ◽  
The Third ◽  
Tensor Square ◽  
Exterior Square

In this paper, in the class of [Formula: see text]-groups of order [Formula: see text], 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 [Formula: see text] of such groups.

scholarly journals The Schur multiplier of groups of order 𝑝5

2019 ◽  
Vol 22 (4) ◽  
pp. 647-687 ◽  
Author(s):  
Sumana Hatui ◽  
Vipul Kakkar ◽  
Manoj K. Yadav
Keyword(s):  
Schur Multiplier ◽  
Tensor Square ◽  
Exterior Square

AbstractIn this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian p-groups of order {p^{5}}. As an application, we determine the capability of groups of order {p^{5}}.

scholarly journals On the size of the third homotopy group of the suspension of an Eilenberg--MacLane space

2014 ◽  
Vol 38 ◽  
pp. 664-671 ◽  
Author(s):  
Peyman NIROOMAND ◽  
Francesco G. RUSSO
Keyword(s):  
Homotopy Group ◽  
The Third

scholarly journals Certain functors of nilpotent Lie algebras with the derived subalgebra of dimension two

2019 ◽  
Vol 19 (01) ◽  
pp. 2050012
Author(s):  
Farangis Johari ◽  
Peyman Niroomand
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Tensor Square ◽  
Exterior Square

By considering the nilpotent Lie algebra with the derived subalgebra of dimension [Formula: see text], we compute some functors including the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.

scholarly journals The third homotopy group as a $$\pi _1$$ π 1 -module

2015 ◽  
Vol 26 (1-2) ◽  
pp. 165-189
Author(s):  
Hans-Joachim Baues ◽  
Beatrice Bleile
Keyword(s):  
Homotopy Group ◽  
The Third

scholarly journals A polycyclic presentation for the 𝑞-tensor square of a polycyclic group

2020 ◽  
Vol 23 (1) ◽  
pp. 97-120
Author(s):  
Ivonildes Ribeiro Martins Dias ◽  
Noraí Romeu Rocco
Keyword(s):  
Homology Group ◽  
Negative Integer ◽  
Polycyclic Group ◽  
Tensor Square ◽  
Exterior Square

AbstractLet G be a group and q a non-negative integer. We denote by {\nu^{q}(G)} a certain extension of the q-tensor square {G\otimes^{q}G} by {G\times G}. In this paper, we describe an algorithm for deriving a polycyclic presentation for {G\otimes^{q}G} when G is polycyclic, via its embedding into {\nu^{q}(G)}. Furthermore, we derive polycyclic presentations for the q-exterior square {G\wedge^{q}G} and for the second homology group {H_{2}(G,\mathbb{Z}_{q})}. Additionally, we establish a criterion for computing the q-exterior center {Z_{q}^{\wedge}(G)} of a polycyclic group G, which is helpful for deciding whether or not G is capable modulo q. These results extend to all {q\geq 0} generalizing methods due to Eick and Nickel for the case {q=0}.

scholarly journals INTERSECTION OF SUBGROUPS IN FREE GROUPS AND HOMOTOPY GROUPS

2008 ◽  
Vol 18 (05) ◽  
pp. 803-823 ◽  
Author(s):  
HANS-JOACHIM BAUES ◽  
ROMAN MIKHAILOV
Keyword(s):  
Abelian Group ◽  
Free Group ◽  
Homotopy Group ◽  
Homotopy Groups ◽  
Natural Homomorphism ◽  
Two Dimensional ◽  
The Third ◽  
Cw Complex ◽  
Intersection Of Subgroups ◽  
The Right

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group π3. This generalizes a result of Gutierrez–Ratcliffe who relate the intersection of two subgroups with the computation of π2. Let K be a two-dimensional CW-complex with subcomplexes K1, K2, K3 such that K = K1 ∪ K2 ∪ K3 and K1 ∩ K2 ∩ K3 is the 1-skeleton K1 of K. We construct a natural homomorphism of π1(K)-modules [Formula: see text] where Ri = ker {π1(K1) → π1(Ki)}, i = 1,2,3 and the action of π1(K) = F/R1R2R3 on the right-hand abelian group is defined via conjugation in F. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.

scholarly journals On the third homotopy group of Orr’s space

2018 ◽  
Vol 18 (1) ◽  
pp. 569-582
Author(s):  
Emmanuel Dror Farjoun ◽  
Roman Mikhailov
Keyword(s):  
Homotopy Group ◽  
The Third

scholarly journals On Pierce-like idempotents and Hopf invariants

2003 ◽  
Vol 2003 (62) ◽  
pp. 3903-3920
Author(s):  
Giora Dula ◽  
Peter Hilton
Keyword(s):  
Homotopy Group ◽  
Hopf Invariant ◽  
The Third ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Matrix Calculus ◽  
Hopf Invariants

Given a setKwith cardinality‖K‖ =n, a wedge decomposition of a spaceYindexed byK, and a cogroupA, the homotopy groupG=[A,Y]is shown, by using Pierce-like idempotents, to have a direct sum decomposition indexed byP(K)−{ϕ}which is strictly functorial ifGis abelian. Given a classρ:X→Y, there is a Hopf invariantHIρon[A,Y]which extends Hopf's definition whenρis a comultiplication. ThenHI=HIρis a functorial sum ofHILoverL⊂K,‖L‖ ≥2. EachHILis a functorial composition of four functors, the first depending only onAn+1, the second only ond, the third only onρ, and the fourth only onYn. There is a connection here with Selick and Walker's work, and with the Hilton matrix calculus, as described by Bokor (1991).

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