A Note on Generalized Direct Products of Groups

1973 ◽  
Vol 25 (1) ◽  
pp. 115-116
Author(s):  
Marlene Schick
Keyword(s):  
Direct Product ◽  
Direct Products ◽  
Compatibility Conditions ◽  
Cyclic Groups ◽  
Products Of Groups ◽  
Finite Set

In [1] Tang proved that the generalized direct product of a finite set of cyclic groups amalgamating subgroups which satisfy certain compatibility conditions always exists. In the proof, Theorem 4.1 is made use of. However, this theorem is not correct since we can construct examples of groups which satisfy the conditions of Theorem 4.1, but whose generalized direct product does not exist. Therefore, a modification of this result as pointed out by Professor Tang is given here, together with the resulting modification of the proof of the result stated above.

scholarly journals Subgroups of direct products closely approximated by direct sums

2017 ◽  
Vol 29 (5) ◽  
pp. 1125-1144 ◽  
Author(s):  
Maria Ferrer ◽  
Salvador Hernández ◽  
Dmitri Shakhmatov
Keyword(s):  
Direct Product ◽  
Coding Theory ◽  
List Type ◽  
Direct Products ◽  
Topological Groups ◽  
Product Topology ◽  
Direct Sums ◽  
Cyclic Groups ◽  
Finite Set ◽  
Infinite Set

AbstractLet I be an infinite set, let {\{G_{i}:i\in I\}} be a family of (topological) groups and let {G=\prod_{i\in I}G_{i}} be its direct product. For {J\subseteq I}, {p_{J}:G\to\prod_{j\in J}G_{j}} denotes the projection. We say that a subgroup H of G is(i)uniformly controllable in G provided that for every finite set {J\subseteq I} there exists a finite set {K\subseteq I} such that {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in K}G_{i})}, (ii)controllable in G provided that {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in I}G_{i})} for every finite set {J\subseteq I},(iii)weakly controllable in G if {H\cap\bigoplus_{i\in I}G_{i}} is dense in H, when G is equipped with the Tychonoff product topology.One easily proves that (i) {\Rightarrow} (ii) {\Rightarrow} (iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrow can be reversed when H is compact, but the second arrow cannot be reversed even when H is compact. Both arrows can be reversed if all groups {G_{i}} are finite. When {G_{i}=A} for all {i\in I}, where A is an abelian group, we show that the first arrow can be reversed for all subgroups H of G if and only if A is finitely generated. We also describe compact groups topologically isomorphic to a direct product of countably many cyclic groups. Connections with coding theory are highlighted.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Presentations for some direct products of groups

1983 ◽  
Vol 28 (1) ◽  
pp. 131-133 ◽  
Author(s):  
P.E. Kenne
Keyword(s):  
Direct Product ◽  
Alternating Group ◽  
Direct Products ◽  
Icosahedral Group ◽  
Products Of Groups

We give efficient presentations for the direct product of two copies of the alternating group of degree five and the direct product of the alternating group of degree five and the binary icosahedral group.

scholarly journals Isomorphisms of Direct Products of Finite Commutative Groups

2013 ◽  
Vol 21 (1) ◽  
pp. 65-74
Author(s):  
Hiroyuki Okazaki ◽  
Hiroshi Yamazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Commutative Group ◽  
Direct Products ◽  
Cyclic Groups ◽  
Composite Number

Summary We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups

scholarly journals MENGKONSTRUKSI DIRECT PRODUCT NEAR RING DAN SMARANDACHE NEAR RING

2019 ◽  
Vol 2 (2) ◽  
pp. 70
Author(s):  
Rizky Muhammad Bagas ◽  
Titi Udjiani SRRM ◽  
Harjito Harjito
Keyword(s):  
Direct Product ◽  
Algebraic Structure ◽  
Cartesian Product ◽  
Direct Products ◽  
Cartesian Products ◽  
Binary Operations ◽  
Products Of Groups

If we have two arbitrary non empty sets  ,then their cartesian product can be constructed. Cartesian products of two sets can be generalized into  number of  sets. It has been found that if the algebraic structure of groups and rings are seen as any set, then the phenomenon of cartesian products of  sets  can be extended to groups and rings. Direct products of groups and rings can be obtained by adding binary operations to the cartesian product. This paper answers the question of whether the direct product phenomenon of groups and rings can also be extended at the  near ring and Smarandache near ring ?. The method in this paper is  by following the method in groups and rings, namely by seen that  near ring and Smarandache near ring  as a set and then build their cartesian products. Next,  the binary operations is adding to the cartesian  products that have been obtained to build the direct product definitions of near ring and near ring Smarandache.

scholarly journals Subnormal subgroups in direct products of groups

1987 ◽  
Vol 42 (2) ◽  
pp. 147-172 ◽  
Author(s):  
Peter Hauck
Keyword(s):  
Lie Algebras ◽  
Direct Product ◽  
Main Part ◽  
Direct Products ◽  
Products Of Groups ◽  
Special Cases ◽  
Associative Rings ◽  
Subnormal Subgroups

AbstractA group G is called normally (subnormally) detectable if the only normal (subnormal) subgroups in any direct product G1 × … × Gn of copies of G are just the direct factors Gi. We give an internal characterization of finite subnormally detectable groups and obtain analogous results for associative rings and for Lie algebras. The main part of the paper deals with a study of normally detectable groups, where we verify a conjecture of T. O. Hawkes in a number of special cases.

scholarly journals Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

2013 ◽  
Vol 21 (3) ◽  
pp. 207-211
Author(s):  
Hiroshi Yamazaki ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Direct Products ◽  
Cyclic Groups

Summary In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

Subgroups of direct products of groups invariant under the action of permutations on factors

2020 ◽  
Vol 30 (4) ◽  
pp. 243-255
Author(s):  
Dmitry A. Burov
Keyword(s):  
Finite Field ◽  
Finite Number ◽  
Vector Space ◽  
Direct Product ◽  
Subdirect Product ◽  
Additive Group ◽  
Direct Products ◽  
Products Of Groups ◽  
Characteristic 2 ◽  
Additional Constraints

AbstractWe study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.

scholarly journals Isomorphisms of Direct Products of Finite Cyclic Groups

2012 ◽  
Vol 20 (4) ◽  
pp. 343-347
Author(s):  
Kenichi Arai ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Fundamental Theorem ◽  
Chinese Remainder Theorem ◽  
Abelian Groups ◽  
Finite Sequence ◽  
Direct Products ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Relative Prime

Summary In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

TEST ELEMENTS IN DIRECT PRODUCTS OF GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 751-756 ◽  
Author(s):  
JOHN C. O'NEILL ◽  
EDWARD C. TURNER
Keyword(s):  
Direct Product ◽  
Commutator Subgroup ◽  
Free Groups ◽  
Surface Groups ◽  
Direct Products ◽  
Products Of Groups ◽  
Test Elements ◽  
Torsion Free

We characterize test elements in the commutator subgroup of a direct product of certain groups in terms of test elements of the factors. This provides explicit examples of test elements in direct products whose factors are free groups or surface groups and a tool for doing the same for torsion free hyperbolic factors.

Sign in / Sign up

Export Citation Format

Share Document