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.

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 Counting Compositions over Finite Abelian Groups

10.37236/7591 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Zhicheng Gao ◽  
Andrew MacFie ◽  
Qiang Wang
Keyword(s):  
Transfer Matrix ◽  
Finite Fields ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Abelian Groups ◽  
Type I ◽  
Type Ii ◽  
Finite Abelian Groups ◽  
Frobenius Theorem ◽  
Exact Counting

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields can be converted to type (i) compositions, whereas Carlitz and "locally Mullen" compositions can be formulated as type (ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections involving certain restricted classes of compositions.

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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