Irreducible products of characters

2019 ◽  
Vol 19 (04) ◽  
pp. 2050069
Author(s):  
Huimin Chang ◽  
Ping Jin
Keyword(s):  
Finite Groups ◽  
Solvable Groups ◽  
Primitive Character ◽  
Unique Factorization

We introduce the notion of Fitting characters for arbitrary finite groups, and prove that under some conditions the product of these characters is irreducible and the unique factorization of this form also holds. Moreover, we show that any nonlinear quasi-primitive character of solvable groups can be uniquely factored (up to multiplication by linear characters) as the product of certain Fitting characters on some extension groups.

scholarly journals A Class of Solvable Groups

1959 ◽  
Vol 11 ◽  
pp. 311-320 ◽  
Author(s):  
Daniel Gorenstein ◽  
I. N. Herstein
Keyword(s):  
Finite Groups ◽  
Prime Order ◽  
Solvable Groups ◽  
Unpublished Paper

Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB, where A and B are subgroups of G. The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B.More recently the structure of finite groups G which have a representation in the form ABA, where A and B are subgroups of G, has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba', where a, a’ are in A, and b ≠ 1 is in B.

Orbits and Stabilizers for Solvable Linear Groups

2006 ◽  
Vol 49 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Irreducible Character ◽  
General Class ◽  
Frobenius Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Linear Groups ◽  
Finite Vector ◽  
Finite Vector Space

AbstractWe extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group G on a finite vector space V under certain conditions, to a more general class of finite solvable groups G. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

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 Zeros of primitive characters of finite groups

2020 ◽  
Vol 23 (2) ◽  
pp. 193-216 ◽  
Author(s):  
Sesuai Yash Madanha
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Solvable Groups ◽  
Classical Theorem ◽  
Complex Character ◽  
Zeros Of Characters

AbstractWe classify finite non-solvable groups with a faithful primitive irreducible complex character that vanishes on a unique conjugacy class. Our results answer a question of Dixon and Rahnamai Barghi and suggest an extension of Burnside’s classical theorem on zeros of characters.

On semidirectly closed non-aperiodic pseudovarieties of finite monoids

2020 ◽  
Vol 63 (4) ◽  
pp. 913-928
Author(s):  
Jiří Kaďourek
Keyword(s):  
Prime Number ◽  
Finite Groups ◽  
Complete Lattice ◽  
Solvable Groups ◽  
Finite Monoids ◽  
The Continuum

AbstractIt is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

scholarly journals Normal complements in finite solvable groups

1974 ◽  
Vol 18 (3) ◽  
pp. 262-264 ◽  
Author(s):  
Donald K. Friesen
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Solvable Groups ◽  
Hall Subgroup

A well known theorem ([1] page 432) in the study of finite groups states that if P is a Sylow p-subgroup of the finite group G, and if P0 is a normal subgroup of P such that whenever two elements, σ and τ, of P are conjugate in G, the cosets σP0 and τP0 are conjugate in P/P0, then there is a normal subgroup K of G such that G = KP and K ∩ P = P0. In this note we will extend this result to allow P to be any Hall subgroup if G is solvable. More precisely, following theorem will be the proved.

On quasi-primitive characters of solvable groups

2019 ◽  
Vol 18 (02) ◽  
pp. 1950038
Author(s):  
Huimin Chang ◽  
Ping Jin
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Solvable Groups ◽  
Primitive Character ◽  
Odd Degree

Let [Formula: see text] be a quasi-primitive character with odd degree, and suppose that [Formula: see text] is a [Formula: see text]-solvable group. Wilde associated to [Formula: see text] a unique conjugacy class of subgroups [Formula: see text] satisfying [Formula: see text]. We construct in this situation a sequence of character pairs [Formula: see text], where [Formula: see text] is quasi-primitive and each [Formula: see text] is uniquely determined up to conjugacy in [Formula: see text], such that [Formula: see text] and [Formula: see text]. Furthermore, we have [Formula: see text] for each [Formula: see text], and in particular [Formula: see text]. We also prove that the subgroups [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text], and thus present a new description for Wilde’s result.

Finite Groups with Some Subgroups of Given Order

2013 ◽  
Vol 20 (03) ◽  
pp. 457-462 ◽  
Author(s):  
Jiangtao Shi ◽  
Cui Zhang ◽  
Dengfeng Liang
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Solvable Groups ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Prime Power Order

Let [Formula: see text] be the class of groups of non-prime-power order or the class of groups of prime-power order. In this paper we give a complete classification of finite non-solvable groups with a quite small number of conjugacy classes of [Formula: see text]-subgroups or classes of [Formula: see text]-subgroups of the same order.

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.

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