scholarly journals The Wielandt subgroup of metacyclic p-groups

1990 ◽  
Vol 42 (3) ◽  
pp. 499-510 ◽  
Author(s):  
Elizabeth A. Ormerod
Keyword(s):  
Finite Group ◽  
Nilpotency Class ◽  
Characteristic Subgroup ◽  
Normal Series ◽  
A Finite Group ◽  
Subnormal Subgroups

The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p–group is identified, and using this information it is shown that if a metacyclic p–group has Wielandt length n, its nilpotency class is n or n + 1.

Generalized Fitting subgroup of a group of finite Morley rank

1991 ◽  
Vol 56 (4) ◽  
pp. 1391-1399 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Morley Rank ◽  
Fitting Subgroup ◽  
Generalized Fitting Subgroup ◽  
Finite Morley Rank ◽  
The Generalized Fitting Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractWe define a characteristic and definable subgroup F*(G) of any group G of finite Morley rank that behaves very much like the generalized Fitting subgroup of a finite group. We also prove that semisimple subnormal subgroups of G are all definable and that there are finitely many of them.

On the generalized norm of a finite group

2015 ◽  
Vol 15 (01) ◽  
pp. 1650008 ◽  
Author(s):  
Lü Gong ◽  
Libo Zhao ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
A Finite Group ◽  
Subnormal Subgroups

The main aim of this paper is to investigate two characteristic subgroups ω𝒜(G) and θ𝒜(G) of a finite group G, which are defined as the intersections of the normalizers of derived subgroups of subnormal and non-subnormal subgroups of G respectively. Our main theory improve and extend some earlier results.

Gradewise Properties of Subgroups and Their Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 255-266
Author(s):  
Wenbin Guo ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Binary Relation ◽  
Finite Groups ◽  
Normal Series ◽  
A Finite Group

For each finite group E, let Θ(E) be a binary relation on the set of all subgroups of E. If A and B are subgroups of a finite group G, then we say that the pair (A, B) enjoys the gradewise property (resp., generalized gradewise property) Θ in G if G has a normal series Γ : 1 = G0 [Formula: see text] such that for each [Formula: see text], we have [Formula: see text] (resp., we have [Formula: see text]. Using these concepts, we obtain some new characterizations for solubility and supersolubility of finite groups and generalize some known results.

A Characteristic Subgroup of a p-Stable Group

1968 ◽  
Vol 20 ◽  
pp. 1101-1135 ◽  
Author(s):  
George Glauberman
Keyword(s):  
Finite Group ◽  
Characteristic Subgroup ◽  
Stable Group ◽  
A Finite Group

Let p be a prime, and let S be a Sylow p-subgroup of a finite group G. J. Thompson (13; 14) has introduced a characteristic subgroup JR(S) and has proved the following results:(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(JR(S)) have normal p-complements.

scholarly journals On Finite Quasi-Core-p p-Groups

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1147
Author(s):  
Jiao Wang ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Nilpotency Class ◽  
Frattini Subgroup ◽  
Normal Core ◽  
A Finite Group

Given a positive integer n, a finite group G is called quasi-core-n if ⟨ x ⟩ / ⟨ x ⟩ G has order at most n for any element x in G, where ⟨ x ⟩ G is the normal core of ⟨ x ⟩ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

On the Intersection of the Normalizers of the Nilpotent Residuals of All Subgroups of a Finite Group

2013 ◽  
Vol 20 (02) ◽  
pp. 349-360 ◽  
Author(s):  
Lü Gong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Characteristic Subgroup ◽  
A Finite Group ◽  
The Relationship

In this paper, a characteristic subgroup [Formula: see text] of a finite group G is defined, which is the intersection of the normalizers of the nilpotent residuals of all subgroups of G, and the properties of [Formula: see text] and the relationship between [Formula: see text] and the group G are investigated.

scholarly journals A NOTE ON p-CENTRAL GROUPS

2013 ◽  
Vol 55 (2) ◽  
pp. 449-456
Author(s):  
RACHEL CAMINA ◽  
ANITHA THILLAISUNDARAM
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Prime Number ◽  
Nilpotency Class ◽  
Schur Multiplier ◽  
Tate Cohomology ◽  
Explicit Bound ◽  
Covering Group ◽  
A Finite Group

AbstractA group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$, where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology Hn(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

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