GROUPS OF FINITE NORMAL LENGTH

2018 ◽  
Vol 97 (2) ◽  
pp. 229-239
Author(s):  
FRANCESCO DE GIOVANNI ◽  
ALESSIO RUSSO
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Commutator Subgroup ◽  
Study Groups ◽  
Normal Closure ◽  
Normal Length ◽  
Sylow Tower

Let $k$ be a nonnegative integer. A subgroup $X$ of a group $G$ has normal length $k$ in $G$ if all chains between $X$ and its normal closure $X^{G}$ have length at most $k$, and $k$ is the length of at least one of these chains. The group $G$ is said to have finite normal length if there is a finite upper bound for the normal lengths of its subgroups. The aim of this paper is to study groups of finite normal length. Among other results, it is proved that if all subgroups of a locally (soluble-by-finite) group $G$ have finite normal length in $G$, then the commutator subgroup $G^{\prime }$ is finite and so $G$ has finite normal length. Special attention is given to the structure of groups of normal length $2$. In particular, it is shown that finite groups with this property admit a Sylow tower.

Finite groups with weakly ℋC-embedded subgroups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050093 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]. We say that [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text]. For each prime [Formula: see text] dividing the order of [Formula: see text] let [Formula: see text] be a Sylow [Formula: see text]-subgroup of [Formula: see text]. We fix a subgroup of [Formula: see text] of order [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that every subgroup of [Formula: see text] of order [Formula: see text] [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.

scholarly journals The Chromatic Number of Finite Group Cayley Tables

10.37236/7874 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Kevin Halasz ◽  
E. S. Mahmoodian
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Chromatic Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Latin Square ◽  
Finite Abelian Groups ◽  
Cayley Table ◽  
Minimum Number ◽  
Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

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.

scholarly journals B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

2006 ◽  
Vol 74 (1) ◽  
pp. 121-132 ◽  
Author(s):  
A. Abdollahi ◽  
A. Azad ◽  
A. Mohammadi Hassanabadi ◽  
M. Zarrin
Keyword(s):  
Finite Group ◽  
Exponential Function ◽  
Finite Groups ◽  
Upper Bound ◽  
Partial Answer ◽  
A Finite Group

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

scholarly journals Automorphisms of Finite Groups and their Fixed-Point Groups

1969 ◽  
Vol 9 (3-4) ◽  
pp. 467-477 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Prime Order ◽  
Soluble Group ◽  
Point Groups ◽  
Derived Series ◽  
A Finite Group

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

On p-supersolvability of finite groups

2015 ◽  
Vol 52 (4) ◽  
pp. 504-510
Author(s):  
Mohamed Asaad
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let HsG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that HG = HT and H ∩ T ≦ HsG, where HG is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.

scholarly journals MAPS OF SURFACE GROUPS TO FINITE GROUPS WITH NO SIMPLE LOOPS IN THE KERNEL

2000 ◽  
Vol 09 (08) ◽  
pp. 1029-1036 ◽  
Author(s):  
CHARLES LIVINGSTON
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Simple Closed Curve ◽  
Orientable Surface ◽  
Closed Curve ◽  
Surface Groups ◽  
Closed Orientable Surface

Let Fg denote the closed orientable surface of genus g. What is the least order finite group, Gg, for which there is a homomorphism ψ:π1(Fg)→Gg so that nontrivial simple closed curve on Fg represents an element in Ker (ψ)? For the torus it is easily seen that G1=Z2×Z2 suffices. We prove here that G2 is a group of order 32 and that an upper bound for the order of Gg is given by g2g+1. The previously known upper bound was greater than 2g22g.

scholarly journals Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

2017 ◽  
Vol 16 (03) ◽  
pp. 1750043
Author(s):  
Martino Garonzi ◽  
Dan Levy ◽  
Attila Maróti ◽  
Iulian I. Simion
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Minimal Length ◽  
Real Constant ◽  
Carter Subgroup ◽  
Positive Real ◽  
Solvable Length ◽  
A Finite Group ◽  
Positive Real Constant

We consider factorizations of a finite group [Formula: see text] into conjugate subgroups, [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of [Formula: see text]. We also show that every solvable group [Formula: see text] is a product of at most [Formula: see text] conjugates of a Carter subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

scholarly journals Finite groups which are the product of two nilpotent subgroups

1973 ◽  
Vol 9 (2) ◽  
pp. 267-274 ◽  
Author(s):  
Fletcher Gross
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Frattini Subgroup ◽  
Derived Length ◽  
A Finite Group

Suppose G = AB where G is a finite group and A and B are nilpotent subgroups. It is proved that the derived length of G modulo its Frattini subgroup is at most the sum of the classes of A and B. An upper bound for the derived length of G in terms of the derived lengths of A and B also is obtained.

On the maximum number of the pairwise noncommuting elements in a finite group

2016 ◽  
Vol 15 (10) ◽  
pp. 1650197 ◽  
Author(s):  
Seyyed Majid Jafarian Amiri ◽  
Halimeh Madadi
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Structural Properties ◽  
Finite Groups ◽  
Upper Bound ◽  
Frobenius Group ◽  
Maximum Size ◽  
Partial Answer ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] be the maximum size of a set of pairwise noncommuting elements in [Formula: see text]. In this paper, we give an upper bound of [Formula: see text] for an arbitrary nilpotent group [Formula: see text]. As an application of this result, we give a partial answer to Question 2.8 of [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000) 139–146]. Also we compute [Formula: see text] when [Formula: see text] is a Frobenius group. Finally we describe structural properties of all groups [Formula: see text] with [Formula: see text].

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