The Homological Functors of Some Abelian Groups of Prime Power Order

2014 ◽  
Vol 71 (5) ◽  
Author(s):  
Rosita Zainal ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Samad Rashid
Keyword(s):  
Tensor Product ◽  
Prime Power ◽  
Homotopy Theory ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Schur Multiplier ◽  
Integer Coefficients ◽  
Nonabelian Tensor ◽  
Tensor Square ◽  
Special Case

The homological functors of a group were first introduced in homotopy theory. Some of the homological functors including the nonabelian tensor square and the Schur multiplier of abelian groups of prime power order are determined in this paper. The nonabelian tensor square of a group G introduced by Brown and Loday in 1987 is a special case of the nonabelian tensor product. Meanwhile, the Schur multiplier of G is the second cohomology with integer coefficients is named after Issai Schur. The aims of this paper are to determine the nonabelian tensor square and the Schur multiplier of abelian groups of order p5, where p is an odd prime

scholarly journals On the non-albelian tensor square of a nilpotent group of class two

1994 ◽  
Vol 36 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Michael R. Bacon
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Homotopy Theory ◽  
Nonabelian Tensor ◽  
Image Position ◽  
Tensor Square ◽  
Special Case

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations,for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].

scholarly journals SOME CONSIDERATIONS ON THE NONABELIAN TENSOR SQUARE OF CRYSTALLOGRAPHIC GROUPS

2011 ◽  
Vol 04 (02) ◽  
pp. 271-282 ◽  
Author(s):  
Ahmad Erfanian ◽  
Francesco G. Russo ◽  
Nor Haniza Sarmin
Keyword(s):  
Tensor Product ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Schur Multiplier ◽  
Crystallographic Groups ◽  
Nonabelian Tensor ◽  
Tensor Square

The nonabelian tensor square G ⊗ G of a polycyclic group G is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of G ⊗ G by looking at that of G, on another hand, we study the nonabelian tensor product of pro–p–groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.

scholarly journals Computing the Nonabelian Tensor Square of Metacyclic p–Groups of Nilpotency Class Two

2013 ◽  
Vol 61 (1) ◽  
Author(s):  
A. M. Basri ◽  
N. H. Sarmin ◽  
N. M. Mohd Ali ◽  
J. R. Beuerle
Keyword(s):  
Commutator Subgroup ◽  
Group Structure ◽  
Nilpotency Class ◽  
Conjugacy Classes ◽  
Schur Multiplier ◽  
Group Order ◽  
Nonabelian Tensor ◽  
Current Article ◽  
Tensor Square ◽  
Compute Structure

In this paper, we develop appropriate programme using Groups, Algorithms and Programming (GAP) software enables performing different computations on various characteristics of all finite nonabelian metacyclic p–groups, p is prime, of nilpotency class 2. Such programme enables to compute structure of the group, order of the group, structure of the center, the number of conjugacy classes, structure of commutator subgroup, abelianization, Whitehead’s universal quadratic functor and other characteristics. In addition, structures of some other groups such as the nonabelian tensor square and various homological functors including Schur multiplier and exterior square can be computed using this programme. Furthermore, by computing the epicenter order or the exterior center order the capability can be determined. In our current article, we only compute the nonabelian tensor square of certain order groups, as an example, and give GAP codes for computing other characteristics and some subgroups.

scholarly journals On Keller's conjecture for certain cyclic groups

1979 ◽  
Vol 22 (1) ◽  
pp. 17-21 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling of Rn by congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.

scholarly journals Lattice embeddings of abelian prime power groups

1997 ◽  
Vol 62 (2) ◽  
pp. 259-278 ◽  
Author(s):  
Roland Schmidt
Keyword(s):  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
Prime Power Order

AbstractWe solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.

Actions of finite abelian groups of odd prime power order

1964 ◽  
pp. 124-146
Author(s):  
P. E. Conner ◽  
E. E. Floyd
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Finite Abelian Groups

scholarly journals Schur multipliers of special 𝑝-groups of rank 2

2020 ◽  
Vol 23 (1) ◽  
pp. 85-95
Author(s):  
Sumana Hatui
Keyword(s):  
Prime Power ◽  
Prime Power Order ◽  
Schur Multiplier ◽  
Schur Multipliers ◽  
Rank 2

AbstractLet G be a special p-group with center of order {p^{2}}. Berkovich and Janko asked to find the Schur multiplier of G in [Y. Berkovich and Z. Janko, Groups of Prime Power Order. Volume 3, De Gruyter Exp. Math. 56, Walter de Gruyter, Berlin, 2011; Problem 2027]. In this article, we answer this question by explicitly computing the Schur multiplier of these groups.

scholarly journals Enumeration of Certain Subgroups of Abelian p-Groups

1962 ◽  
Vol 13 (1) ◽  
pp. 1-4 ◽  
Author(s):  
I. J. Davies
Keyword(s):  
Abelian Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Integer Coefficients

The number of distinct types of Abelian group of prime-power order pn is equal to the number of partitions of n. Let (ρ) = (ρ1, ρ2, …, ρr) be a partition of n and let (μ) = (μ1, μ2, …, μs) be a partition of m, with ρ1≧ρ2≧…≧ρr and μ1≧μ2≧…≧μs, ρi≧μi, r≧s, n>m. The number of subgroups of type μ in an Abelian p-group of type (ρ) is a function of the two partitions (μ) and p, and has been determined as a polynomial in p with integer coefficients by Yeh (1), Delsarte (2) and Kinosita (3). Their results differ in form but are equivalent.

A note on the Schur multiplier of groups of prime power order

2012 ◽  
Vol 61 (2) ◽  
pp. 341-346 ◽  
Author(s):  
Peyman Niroomand
Keyword(s):  
Prime Power ◽  
Prime Power Order ◽  
Schur Multiplier
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