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.

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 NONVANISHING ELEMENTS FOR BRAUER CHARACTERS

2015 ◽  
Vol 102 (1) ◽  
pp. 96-107 ◽  
Author(s):  
SILVIO DOLFI ◽  
EMANUELE PACIFICI ◽  
LUCIA SANUS
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Regular Element ◽  
Solvable Groups ◽  
Vector Spaces ◽  
Brauer Character ◽  
Brauer Characters ◽  
A Finite Group ◽  
Finite Vector Spaces

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

scholarly journals On the kernels of representations of finite groups

1981 ◽  
Vol 22 (2) ◽  
pp. 151-154 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Modular Representation ◽  
Complex Representation ◽  
Representations Of Finite Groups ◽  
A Finite Group

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:(1) G is p-solvable of p-length 1.(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

Planarity of the intersection graph of subgroups of a finite group

2016 ◽  
Vol 15 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Hadi Ahmadi ◽  
Bijan Taeri
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Prime Order ◽  
Undirected Graph ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
A Finite Group

For a nontrivial finite group [Formula: see text] different from a cyclic group of prime order, the intersection graph [Formula: see text] of [Formula: see text] is the simple undirected graph whose vertices are the nontrivial proper subgroups of [Formula: see text] and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper we characterize all finite groups with planar intersection graphs. It turns out that few solvable groups have planar intersection graphs. Also we classify finite groups whose intersection graphs are bipartite, triangle free and forests.

On the number of same order classes of non-abelian subgroups of a finite group

2020 ◽  
pp. 2250036
Author(s):  
Wei Meng ◽  
Guifang Yang ◽  
Jiakuan Lu
Keyword(s):  
Finite Group ◽  
Lower Bounds ◽  
Finite Groups ◽  
Solvable Groups ◽  
Prime Divisors ◽  
Abelian Subgroups ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] denote the set of the prime divisors of [Formula: see text]. The symbol [Formula: see text] denotes the number of same order classes of all non-abelian subgroups of [Formula: see text]. Firstly, the finite groups with [Formula: see text] are classified completely. Secondly, the lower bounds on [Formula: see text] are obtained by the functions of [Formula: see text]. In particular, it is showed that [Formula: see text] for non-solvable groups [Formula: see text]. Finally, the structure of groups with [Formula: see text] is investigated.

scholarly journals Parameterization of Irreducible Characters for p-Solvable Groups

2012 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Outer Automorphisms ◽  
A Finite Group

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

Finite Groups Whose Common-Divisor Graph is Regular

2018 ◽  
Vol 61 (2) ◽  
pp. 329-341
Author(s):  
Mehdi Ghaffarzadeh ◽  
Mohsen Ghasemi ◽  
Mark L. Lewis
Keyword(s):  
Finite Group ◽  
Regular Graph ◽  
Finite Groups ◽  
Complete Graph ◽  
Solvable Groups ◽  
Connected Components ◽  
Irreducible Characters ◽  
Vertex Set ◽  
The Common ◽  
A Finite Group

AbstractLet G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

scholarly journals On the kernels of representations of finite groups II

1990 ◽  
Vol 32 (3) ◽  
pp. 341-347 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Representations Of Finite Groups ◽  
A Finite Group

About fifteen years ago I. M. Isaacs and S. D. Smith [9] gave several character-theoretic characterizations of finite p-solvable groups G with p-length one, where p is a prime number. They proved that for a finite group G with a Sylow p-subgroup P, the following four conditions (a)–(d) are equivalent.

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 Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

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