scholarly journals Octahedron Subgroups and Subrings

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1444
Author(s):  
Jeong-Gon Lee ◽  
Young Bae Jun ◽  
Kul Hur
Keyword(s):  
Normal Subgroup ◽  
Skew Field

In this paper, we define the notions of i-octahedron groupoid and i-OLI [resp., i-ORI and i-OI], and study some of their properties and give some examples. Also we deal with some properties for the image and the preimage of i-octahedron groupoids [resp., i-OLI, i-ORI and i-OI] under a groupoid homomorphism. Next, we introduce the concepts of i-octahedron subgroup and normal subgroup of a group and investigate some of their properties. In particular, we obtain a characterization of an i-octahedron subgroup of a group. Finally, we define an i-octahedron subring [resp., i-OLI, i-ORI and i-OI] of a ring and find some of their properties. In particular, we obtain two characterizations of i-OLI [resp., i-ORI and i-OI] of a ring and a skew field, respectively.

A Useful Characterization of a Normal Subgroup

1979 ◽  
Vol 52 (3) ◽  
pp. 171
Author(s):  
Francis E. Masat
Keyword(s):  
Normal Subgroup

A Useful Characterization of a Normal Subgroup

1979 ◽  
Vol 52 (3) ◽  
pp. 171-173
Author(s):  
Francis E. Masat
Keyword(s):  
Normal Subgroup

A result on p-hypercyclically embedded subgroups

2019 ◽  
Vol 18 (03) ◽  
pp. 1950043
Author(s):  
Changwen Li ◽  
Jianhong Huang ◽  
Bin Hu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
A Finite Group

In this paper, a new characterization of [Formula: see text]-hypercyclical embeddability of a normal subgroup of a finite group is obtained based on the notion of [Formula: see text]-subgroups and some known results are generalized and extended.

Notes on Splitting Extensions of Groups

1968 ◽  
Vol 11 (3) ◽  
pp. 371-374 ◽  
Author(s):  
C.Y. Tang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Similar Type ◽  
Frattini Subgroup ◽  
Abelian Normal Subgroup ◽  
A Finite Group

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

A Characterization of the Commutator Subgroup of a Group

1976 ◽  
Vol 19 (1) ◽  
pp. 93-94 ◽  
Author(s):  
G. Thierrin
Keyword(s):  
Normal Subgroup ◽  
Commutator Subgroup

AbstractAn element a of a semigroup S is n-potent if there exist a1, a2,..., ak∈S such that a = a1a2...ak and If S is a group, the set of n-potent elements is a normal subgroup of S and the set of 1-potent elements is the commutator subgroup of S.

A characterization of hypercyclically embedded subgroups using cover-avoidance property

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Ning Su ◽  
Yanming Wang ◽  
Yangming Li
Keyword(s):  
Normal Subgroup

Abstract.A normal subgroup

scholarly journals Some New Applications of Weakly H-Embedded Subgroups of Finite Groups

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 158
Author(s):  
Li Zhang ◽  
Li-Jun Huo ◽  
Jia-Bao Liu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Normal Closure ◽  
A Finite Group ◽  
New Applications ◽  
New Criterion

A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G, and H ( G ) is the set of all H -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G. As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “ N G ( P ) is p-nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S-quasinormal in G, which means that H T permutes with every Sylow subgroup of G.

First-order characterization of the radical of a finite group

2009 ◽  
Vol 74 (4) ◽  
pp. 1429-1435 ◽  
Author(s):  
John S. Wilson
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
First Order ◽  
Order Language ◽  
A Finite Group

AbstractIt is shown that there is a formula σ(g) in the first-order language of group theory with the following property: for every finite group G, the largest soluble normal subgroup of G consists precisely of the elements g of G such that σ(g) holds.

On the law of nullity

1982 ◽  
Vol 91 (3) ◽  
pp. 357-374 ◽  
Author(s):  
P. M. Cohn ◽  
A. H. Schofield
Keyword(s):  
Global Dimension ◽  
The Other ◽  
Rank Function ◽  
Projective Modules ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Skew Field ◽  
Other Hand ◽  
Semihereditary Rings

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.

scholarly journals DENSELY ORDERED BRAID SUBGROUPS

2007 ◽  
Vol 16 (07) ◽  
pp. 869-877 ◽  
Author(s):  
ADAM CLAY ◽  
DALE ROLFSEN
Keyword(s):  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Braid Groups ◽  
Burau Representation ◽  
Positive Elements

Dehornoy showed that the Artin braid groups Bn are left-orderable. This ordering is discrete, but we show that, for n > 2 the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those n for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of Bn which is discretely ordered by the Dehornoy ordering.

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