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.

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.

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

scholarly journals On the Frattini subgroup of a finite group

2017 ◽  
Vol 470 ◽  
pp. 254-262 ◽  
Author(s):  
Stefanos Aivazidis ◽  
Adolfo Ballester-Bolinches
Keyword(s):  
Finite Group ◽  
Frattini Subgroup ◽  
A Finite Group

scholarly journals Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

BOUNDING THE ORDER OF THE NILPOTENT RESIDUAL OF A FINITE GROUP

2016 ◽  
Vol 94 (2) ◽  
pp. 273-277
Author(s):  
AGENOR FREITAS DE ANDRADE ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Nilpotent Residual ◽  
A Finite Group

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

scholarly journals Almost Engel finite and profinite groups

2016 ◽  
Vol 26 (05) ◽  
pp. 973-983 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Profinite Group ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Nilpotent Residual ◽  
A Finite Group

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

The Structure of G/ Φ (G) in Terms of σ (G)

2006 ◽  
Vol 13 (01) ◽  
pp. 1-8
Author(s):  
Alireza Jamali ◽  
Hamid Mousavi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Group ◽  
Maximal Subgroups ◽  
Finite Soluble Group ◽  
A Finite Group

Let G be a finite group. We let [Formula: see text] and σ (G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/ Φ (G) when G is a finite soluble group with [Formula: see text].

scholarly journals Sums of element orders in groups of odd order

2021 ◽  
pp. 1-15
Author(s):  
Marcel Herzog ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cyclic Group ◽  
Element Orders ◽  
Maximum Value ◽  
Odd Order ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] denote the sum of element orders of [Formula: see text]. If [Formula: see text] is a positive integer let [Formula: see text] be the cyclic group of order [Formula: see text]. It is known that [Formula: see text] is the maximum value of [Formula: see text] on the set of groups of order [Formula: see text], and [Formula: see text] if and only if [Formula: see text] is cyclic of order [Formula: see text]. In this paper, we investigate the second largest value of [Formula: see text] on the set of groups of order [Formula: see text] and the structure of groups [Formula: see text] of order [Formula: see text] with this value of [Formula: see text] when [Formula: see text] is odd.

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