A Useful Characterization of a Normal Subgroup

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

A Useful Characterization of a Normal Subgroup

1979 ◽  
Vol 52 (3) ◽  
pp. 171
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.

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.

A note on automorphism groups of symmetric cubic graphs

2020 ◽  
pp. 2250018
Author(s):  
Jicheng Ma
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Simple Group ◽  
Minimal Normal Subgroup ◽  
Automorphism Groups ◽  
Cubic Graphs ◽  
Normal Subgroups ◽  
Interesting Consequence ◽  
Cubic Graph

We study [Formula: see text]-arc-transitive cubic graph [Formula: see text], and give a characterization of minimal normal subgroups of the automorphism group. In particular, each [Formula: see text] with quasi-primitive automorphism group is characterized. An interesting consequence of this characterization states that a non-solvable minimal normal subgroup [Formula: see text] contains at most 2 copies of non-abelian simple group when it acts transitively on arcs, or contains at most 6 copies of non-abelian simple group when it acts regularly on vertices.

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.

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