scholarly journals Fusion systems and group actions with abelian isotropy subgroups

2013 ◽  
Vol 56 (3) ◽  
pp. 873-886 ◽  
Author(s):  
Özgün Ünlü ◽  
Ergün Yalçin
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Finite Solvable Group ◽  
Recursive Method ◽  
Fusion Systems ◽  
Product Of Spheres ◽  
A Finite Group ◽  
Positive Integers

AbstractWe prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × $\mathbb{S}^{n_1}\$ × … × $\mathbb{S}^{n_k}$ for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

scholarly journals Frobenius action on Carter subgroups

2020 ◽  
Vol 30 (05) ◽  
pp. 1073-1080
Author(s):  
Güli̇n Ercan ◽  
İsmai̇l Ş. Güloğlu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Semidirect Product ◽  
Frobenius Group ◽  
Product Formula ◽  
Finite Solvable Group ◽  
Carter Subgroup ◽  
Text Length ◽  
Fitting Height ◽  
A Finite Group

Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.

scholarly journals ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

2010 ◽  
Vol 81 (2) ◽  
pp. 317-328 ◽  
Author(s):  
MARCEL HERZOG ◽  
PATRIZIA LONGOBARDI ◽  
MERCEDE MAJ
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Connected Graph ◽  
Solvable Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Finitely Generated ◽  
Infinite Case ◽  
Locally Graded Group ◽  
A Finite Group

AbstractLet G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

scholarly journals Non-Abelian finite groups whose character sums are invariant but are not Cayley isomorphism

2019 ◽  
Vol 18 (01) ◽  
pp. 1950013
Author(s):  
Alireza Abdollahi ◽  
Maysam Zallaghi
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Finite Groups ◽  
Closed Subset ◽  
Cayley Graphs ◽  
Character Sums ◽  
Main Question ◽  
Vertex Set ◽  
A Finite Group ◽  
Positive Integers

Let [Formula: see text] be a group and [Formula: see text] an inverse closed subset of [Formula: see text]. By a Cayley graph [Formula: see text], we mean the graph whose vertex set is the set of elements of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. A group [Formula: see text] is called a CI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for some automorphism [Formula: see text] of [Formula: see text]. A finite group [Formula: see text] is called a BI-group if [Formula: see text] for some inverse closed subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then [Formula: see text] for all positive integers [Formula: see text], where [Formula: see text] denotes the set [Formula: see text]. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180–189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159–5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30.

scholarly journals COMMUTATIVITY DEGREE OF A CLASS OF FINITE GROUPS AND CONSEQUENCES

2013 ◽  
Vol 88 (3) ◽  
pp. 448-452 ◽  
Author(s):  
RAJAT KANTI NATH
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Finite Abelian Groups ◽  
A Finite Group ◽  
Open Question

AbstractThe commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

scholarly journals Generalized bases of finite groups

2021 ◽  
Author(s):  
Benjamin Sambale
Keyword(s):  
Finite Group ◽  
Permutation Group ◽  
Finite Groups ◽  
Solvable Groups ◽  
Fusion Systems ◽  
Local Invariant ◽  
Minimal Base ◽  
A Finite Group

AbstractMotivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset $$\Delta $$ Δ of a finite group G is called a p-base (where p is a prime) if $$\langle \Delta \rangle $$ ⟨ Δ ⟩ is a p-group and $$\mathrm {C}_G(\Delta )$$ C G ( Δ ) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.

On the Largest Irreducible Character Degree of a Finite Solvable Group

2010 ◽  
Vol 17 (spec01) ◽  
pp. 925-927 ◽  
Author(s):  
M. H. Jafari
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Irreducible Character ◽  
Prime Order ◽  
Character Degree ◽  
Finite Solvable Group ◽  
A Finite Group ◽  
Irreducible Character Degree

Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn.

Finite groups with some s-conditionally permutable subgroups

2017 ◽  
Vol 16 (12) ◽  
pp. 1750224
Author(s):  
S. E. Mirdamadi ◽  
G. R. Rezaeezadeh
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be [Formula: see text]-conditionally permutable in [Formula: see text] if for every Sylow subgroup [Formula: see text] of [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text]. In this paper, the structure of solvable group [Formula: see text] in which every [Formula: see text]-subgroup of [Formula: see text] or every subnormal subgroup of [Formula: see text] is [Formula: see text]-conditionally permutable in [Formula: see text] is described. Let [Formula: see text] be a solvable group and [Formula: see text] the largest prime dividing [Formula: see text]. Suppose further that [Formula: see text] is the Sylow [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text]. We are going to show that [Formula: see text] is a PST-group if and only if every subnormal subgroup of [Formula: see text] is [Formula: see text]-conditionally permutable in [Formula: see text].

A Note on the p-Deficiency Class of a Finite Group

2012 ◽  
Vol 19 (02) ◽  
pp. 353-358 ◽  
Author(s):  
Tianze Li ◽  
Weigang Xu ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Characteristic P ◽  
P Deficiency ◽  
Class 1 ◽  
A Finite Group ◽  
P Type ◽  
The Relationship

In this note, we explore the relationship between finite groups of characteristic p type and those of p-deficiency class 1. We study the structure of finite groups of characteristic p type. Besides, we show that the p-rank (resp., p-length) of a p-solvable group which is of exact p-deficiency class r(> 0) is bounded by r (resp., a function of r).

Lower bounds on conjugacy classes of non-nilpotent subgroups in a finite group

2014 ◽  
Vol 14 (03) ◽  
pp. 1550039 ◽  
Author(s):  
Wei Meng ◽  
Jiakuan Lu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Lower Bounds ◽  
Conjugacy Classes ◽  
Finite Solvable Group ◽  
Prime Divisors ◽  
A Finite Group

For a finite group G, let γ(G) denote the number of conjugacy classes of all non-nilpotent subgroups of G, and let π(G) denote the set of the prime divisors of |G|. In this paper, we establish lower bounds on γ(G). In fact, we show that if G is a finite solvable group, then γ(G) = 0 or γ(G) ≥ 2|π(G)|-2, and if G is non-solvable, then γ(G) ≥ |π(G)| + 1. Both lower bounds are best possible.

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