Sylow Subgroups in the Group of Isomorphisms of Prime Power Abelian Groups

1923 ◽  
Vol 45 (4) ◽  
pp. 223 ◽  
Author(s):  
Harry Albert Bender
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Sylow 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 On the Cayley Isomorphism Problem for Cayley Objects of Nilpotent Groups of Some Orders

10.37236/3123 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Edward Dobson
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Necessary Condition ◽  
Sylow Subgroups ◽  
Arithmetic Properties

We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group $G$ whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of $G$ in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that ${\mathbb Z}_q\times{\mathbb Z}_p^2\times{\mathbb Z}_m$ is a CI-group with respect to digraphs, where $q$ and $p$ are primes with $p^2 < q$ and $m$ is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on $q$ and $p$).

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.

scholarly journals On efficient computation of sums of characters on the basis of A. G. Postnikov methods

2021 ◽  
pp. 13-16
Author(s):  
Nikolaj Glazunov
Keyword(s):  
Finite Fields ◽  
Prime Power ◽  
Abelian Groups ◽  
Summation Method ◽  
Kloosterman Sums ◽  
Efficient Computation ◽  
Finite Abelian Groups

An efficient p-adic method and the structure of an algorithm for computing the sums of characters of finite abelian groups are presented. The method and algorithm are based on the A.G. Postnikov summation method of characters modulo a prime power and its developments. A brief survey of the theory of characters of finite abelian groups, p-adic arithmetic and analysis is presented. Questions of the efficiency of p-adic methods are discussed. Moreover, we present results of computation of other types of sums of characters (Kloosterman sums), which are connecting with Artin-Schreier coverings over prime finite fields. The corresponding method and algorithm are based on the development of another method by A.G. Postnikov. Examples of computation of sums of characters are given.

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 On varieties of metabelian groups of prime-power exponent

1972 ◽  
Vol 14 (2) ◽  
pp. 129-154 ◽  
Author(s):  
M. S. Brooks
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Product Variety ◽  
Power Exponent ◽  
Metabelian Groups ◽  
Proper Subvariety

Let Un denote the variety of abelian groups of exponent dividing n, and let p be an arbitrary prime. In this paper all non-nilpotent, join-ireducible subvarieties of the product variety UpUp2 are determined. The proper subvarieties of this kind in fact form an infinite ascending chain …, and an arbitrary proper subvariety B of UpUp2 is either nilpotent or a join , where L is nilpotent and k is uniquely determined by B.

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 Nilpotent groups are not dualizable

2002 ◽  
Vol 72 (2) ◽  
pp. 173-180 ◽  
Author(s):  
R. Quackenbush ◽  
C. S. Szabó
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Nilpotent Subgroup ◽  
Pontryagin Duality ◽  
Sylow Subgroups ◽  
Companion Article

AbstractIt is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.

Prime-Power Abelian Groups Generated by a Set of Conjugates Under a Special Automorphism

1933 ◽  
Vol 55 (1/4) ◽  
pp. 553
Author(s):  
H. R. Brahana
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Special Automorphism
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