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.

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.

Products of two-sorted structures

1972 ◽  
Vol 37 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Finite Number ◽  
Direct Product ◽  
Direct Products ◽  
Direct Sums ◽  
First Order ◽  
Order Language ◽  
Universal Equivalence ◽  
Relational Systems ◽  
Multiple Operation ◽  
Definition Of

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

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 Subgroups of 3-Factor Direct Products

2019 ◽  
Vol 73 (1) ◽  
pp. 19-38
Author(s):  
Daniel Neuen ◽  
Pascal Schweitzer
Keyword(s):  
Explicit Formula ◽  
Direct Product ◽  
Subdirect Product ◽  
Structure Theorem ◽  
Symmetric Groups ◽  
Direct Products ◽  
Special Cases ◽  
Subgroup Lattices ◽  
Subdirect Products

Abstract Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric 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 Abelian 2-subgroups of finite symplectic groups in characteristic 2

1982 ◽  
Vol 33 (3) ◽  
pp. 345-350 ◽  
Author(s):  
M. J. J. Barry ◽  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Preceding Paper ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Characteristic 2 ◽  
Finite Symplectic Groups

AbstractIf Sp(V) is the symplectic group of a vector space V over a finite field of characteristic p, and r is a positive integer, the abelian p-subgroups of largest order in Sp(V) whose fixed subspaces in V have dimension at least r were determined in the preceding paper, in the case p ≠ 2. Here we deal with the case p = 2. Our results also complete earlier work on the orthogonal 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.

scholarly journals Irreducible subgroups of symplectic groups in characteristic 2

2002 ◽  
Vol 73 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Christopher Parker ◽  
Peter Rowley
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Symplectic Groups ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
Zero Vector

AbstractSuppose that V is a finite dimensional vector space over a finite field of characteristic 2, G is the symplectic group on V and a is a non-zero vector of V. Here we classify irreducible subgroups of G containing a certain subgroup of O2(StabG(a)) all of whose non-trivial elements are 2-transvections.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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