scholarly journals Construction of 2-Gyrogroups in Which Every Proper Subgyrogroup Is Either a Cyclic or a Dihedral Group

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 316
Author(s):  
Soheila Mahdavi ◽  
Ali Reza Ashrafi ◽  
Mohammad Ali Salahshour ◽  
Abraham Albert Ungar
Keyword(s):  
Normal Subgroup ◽  
Dihedral Group ◽  
Subgroup Lattice

In this paper, a 2-gyrogroup G(n) of order 2n, n≥3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup lattice of G(n) are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order 2n, which causes us to use the name dihedral gyrogroup for this class of gyrogroups of order 2n. Moreover, all proper subgyrogroups of G(n) are subgroups.

On the automorphism group of the integral group ring of the infinite dihedral group

1988 ◽  
Vol 108 (1-2) ◽  
pp. 1-10
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Group Ring ◽  
Polynomial Ring ◽  
Dihedral Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Index 2 ◽  
Skolem Theorem ◽  
Inner Automorphisms

SynopsisLet ℤ(G) be the integral group ring of the infinite dihedral groupG; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically intoM2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t]From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

scholarly journals On some classes of sublattices of the subgroup lattice

2019 ◽  
pp. 35-47 ◽  
Author(s):  
Alexander N. Skiba
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
Subgroup Lattice ◽  
Normal Closure ◽  
Normal Subgroups ◽  
Normal Section ◽  
Normal Core

In this paper G always denotes a group. If K and H are subgroups of G, where K is a normal subgroup of H, then the factor group of H by K is called a section of G. Such a section is called normal, if K and H are normal subgroups of G, and trivial, if K and H are equal. We call any set S of normal sections of G a stratification of G, if S contains every trivial normal section of G, and we say that a stratification S of G is G-closed, if S contains every such a normal section of G, which is G-isomorphic to some normal section of G belonging S. Now let S be any G-closed stratification of G, and let L be the set of all subgroups A of G such that the factor group of V by W, where V is the normal closure of A in G and W is the normal core of A in G, belongs to S. In this paper we describe the conditions on S under which the set L is a sublattice of the lattice of all subgroups of G and we also discuss some applications of this sublattice in the theory of generalized finite T-groups.

scholarly journals The Wielandt subgroup of a polycyclic group

1991 ◽  
Vol 33 (2) ◽  
pp. 231-234 ◽  
Author(s):  
John Cossey
Keyword(s):  
Normal Subgroup ◽  
Dihedral Group ◽  
Minimal Normal Subgroup ◽  
Polycyclic Group ◽  
Minimum Condition ◽  
Basic Properties ◽  
Subnormal Subgroups

The purpose of this paper is to establish some basic properties of the Wielandt subgroup of a polycyclic group. The Wielandt subgroup of a group G is defined to be the intersection of the normalisers of all the subnormal subgroups of G and is denoted by ω(G). In 1958 Wielandt [9] showed that any minimal normal subgroup with the minimum condition on subnormal subgroups is contained in the Wielandt subgroup: it follows that the Wielandt subgroup of an artinian group is nontrivial. In contrast, the Wielandt subgroup of a polycyclic group can be trivial; an easy example is given by the infinite dihedral group. We will show that the Wielandt subgroup of a polycyclic group is close to being central.

scholarly journals The Non-Normal Subgroup Graph of Some Dihedral Groups

2020 ◽  
pp. 1-7
Author(s):  
Nabilah Fikriah Rahin ◽  
Nor Haniza Sarmin ◽  
Sheila Ilangovan
Keyword(s):  
Normal Subgroup ◽  
Dihedral Group ◽  
Simple Graph ◽  
Dihedral Groups ◽  
Vertex Set

Let G be a finite groupand H is a subgroup of G. The subgroup graph of H in G is defined as a directed simple graph with vertex set G and two distinct elements x and y are adjacent if and only if xy∈H. In this paper, the work on subgroup graph is extended by defining a new graph called the non-normal subgroup graph. The subgroup graph is determined for some dihedral groupsof order 2n when the subgroup is non-normal. Keywords: subgroup graph; non-normal subgroup; dihedral group

Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

2016 ◽  
Vol 23 (03) ◽  
pp. 409-422 ◽  
Author(s):  
Vipul Kakkar ◽  
R. P. Shukla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Dihedral Group ◽  
Elementary Proof ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Right Loop

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this paper, we give some characterizations for the normality of H in G. As a consequence we get a very short and elementary proof of the main theorem of a paper of Lal and Shukla, which avoids the use of the classification of finite simple groups. Further, we study the isotopy between the transversals in some groups and determine the number of isotopy classes of transversals of a subgroup of order 2 in D2p, the dihedral group of order 2p, where p is an odd prime and the isotopism classes are formed with respect to induced right loop structures.

scholarly journals Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group

2011 ◽  
Vol 27 (2) ◽  
pp. 193-199
Author(s):  
CAROLINA CONTIU ◽  
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Sufficient Conditions ◽  
Subgroup Lattice ◽  
Necessary And Sufficient Conditions ◽  
Arbitrary Group ◽  
Necessary And Sufficient ◽  
Arbitrary Abelian Group

In this paper, we provide necessary and sufficient conditions under which a lattice is isomorphic to the subgroup lattice of an arbitrary abelian group. We also give necessary and sufficient conditions for a lattice L, to be isomorphic to the normal subgroup lattice of an arbitrary group.

Canonical Submersion and Lie Homomorphisms Normal Subgroup

2020 ◽  
Vol 23 (1) ◽  
pp. 97-101
Author(s):  
Mikhail Petrichenko ◽  
Dmitry W. Serow
Keyword(s):  
Normal Subgroup ◽  
Canonical System ◽  
System Of Equations ◽  
Core Group ◽  
The Core ◽  
Switch Group

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.

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