Finite non-Abelian groups with complemented non-Abelian subgroups

1978 ◽  
Vol 29 (6) ◽  
pp. 541-544 ◽  
Author(s):  
P. P. Baryshovets
Keyword(s):  
Abelian Groups ◽  
Abelian Subgroups

GROUP CODES OVER NON-ABELIAN GROUPS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350037 ◽  
Author(s):  
CRISTINA GARCÍA PILLADO ◽  
SANTOS GONZÁLEZ ◽  
CONSUELO MARTÍNEZ ◽  
VICTOR MARKOV ◽  
ALEXANDER NECHAEV
Keyword(s):  
Finite Group ◽  
Open Problem ◽  
Abelian Groups ◽  
Base Field ◽  
Minimal Order ◽  
Group Codes ◽  
Abelian Subgroups ◽  
A Finite Group ◽  
Del Rio

Let G be a finite group and F a field. We show that all G-codes over F are abelian if the order of G is less than 24, but for F = ℤ5 and G = S4 there exist non-abelian G-codes over F, answering to an open problem posed in [J. J. Bernal, Á. del Río and J. J. Simón, An intrinsical description of group codes, Des. Codes Cryptogr.51(3) (2009) 289–300]. This problem is related to the decomposability of a group as the product of two abelian subgroups. We consider this problem in the case of p-groups, finding the minimal order for which all p-groups of such order are decomposable. Finally, we study if the fact that all G-codes are abelian remains true when the base field is changed.

On Groups Saturated with Abelian Subgroups

1998 ◽  
Vol 08 (04) ◽  
pp. 443-466 ◽  
Author(s):  
Lev S. Kazarin ◽  
Leonid A. Kurdachenko ◽  
Igor Ya. Subbotin
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Normal Subgroups ◽  
Main Subject ◽  
Maximal Condition ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Locally Nilpotent ◽  
Abelian Subgroups ◽  
Weak Maximal Condition

Groups with the weak maximal condition on non-abelian subgroups are the main subject of this research. Locally finite groups with this property are abelian or Chemikov. Non-abelian groups with the weak maximal condition on non-abelian subgroups, which have an ascending series of normal subgroups with locally nilpotent or locally finite factors, are described in this article.

scholarly journals On maximal abelian groups of maps

1993 ◽  
Vol 55 (3) ◽  
pp. 414-420 ◽  
Author(s):  
Reinhard Winkler
Keyword(s):  
Symmetric Group ◽  
Fixed Points ◽  
Abelian Subgroup ◽  
Abelian Groups ◽  
Simple Description ◽  
Abelian Subgroups

AbstractThe paper gives a rather simple description of all maximal abelian subgroups H of the symmetric group SM acting on an arbitrary set M. In the case of finite M this result is used to determine the maximal cardinality of such an H and the maximal number of permutations without fixed points contained in an abelian subgroup of SM.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

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