scholarly journals Generators of split extensions of Abelian groups by cyclic groups

2018 ◽  
Vol 12 (2) ◽  
pp. 765-802
Author(s):  
Luc Guyot
Keyword(s):  
Abelian Groups ◽  
Cyclic Groups ◽  
Split Extensions

UNITS OF THE GROUP ALGEBRA 𝔽2κ(C2 × D8)

2011 ◽  
Vol 10 (04) ◽  
pp. 643-647 ◽  
Author(s):  
JOE GILDEA
Keyword(s):  
Group Algebra ◽  
Unit Group ◽  
Cyclic Groups ◽  
Split Extensions ◽  
Characteristic 2

The structure of the unit group of the group algebra of the group C2 × D8 over any field of characteristic 2 is established in terms of split extensions of cyclic groups.

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Sur les Sous-Groupes Purifiables des Groupes Abéliens Primaires

1990 ◽  
Vol 33 (1) ◽  
pp. 11-17 ◽  
Author(s):  
K. Benabdallah ◽  
C. Piché
Keyword(s):  
Abelian Groups ◽  
Divisible Group ◽  
Direct Sums ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Direct Sum

AbstractThe class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

The intersection graph of a group

2015 ◽  
Vol 14 (05) ◽  
pp. 1550065 ◽  
Author(s):  
S. Akbari ◽  
F. Heydari ◽  
M. Maghasedi
Keyword(s):  
Disjoint Union ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Free Graph ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
Vertex Set

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

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 The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

scholarly journals HIGHER INDICATORS FOR SOME GROUPS AND THEIR DOUBLES

2012 ◽  
Vol 11 (02) ◽  
pp. 1250030 ◽  
Author(s):  
MARC KEILBERG
Keyword(s):  
Semidirect Product ◽  
Prime Order ◽  
Abelian Groups ◽  
Dihedral Groups ◽  
Cyclic Groups ◽  
Generalized Quaternion ◽  
Drinfel'd Doubles

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups include the dihedral groups, the non-abelian groups of order pq, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.

scholarly journals On nilpotent but not abelian groups and abelian but not cyclic groups

1988 ◽  
Vol 28 (3) ◽  
pp. 363-368 ◽  
Author(s):  
Paul Erdös ◽  
Michael E. Mays
Keyword(s):  
Abelian Groups ◽  
Cyclic Groups
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