A NOTE ON DERIVED LENGTH AND CHARACTER DEGREES

2020 ◽  
pp. 1-7
Author(s):  
BURCU ÇINARCI ◽  
TEMHA ERKOÇ
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Commutator Subgroup ◽  
Sufficient Conditions ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Irreducible Characters ◽  
Derived Length

Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$ . We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

Real characters, monolithic characters and the Taketa inequality

2019 ◽  
Vol 18 (10) ◽  
pp. 1950183 ◽  
Author(s):  
Burcu Çınarcı ◽  
Temha Erkoç
Keyword(s):  
Solvable Group ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Complex Character ◽  
The Real ◽  
Derived Length

In this paper, we prove that the Taketa inequality, namely the derived length of a finite solvable group [Formula: see text] is less than or equal to the number of distinct irreducible complex character degrees of [Formula: see text], is true under some conditions related to the real and the monolithic characters of [Formula: see text].

A Note on Brauer Character Degrees of Solvable Groups

1991 ◽  
Vol 34 (3) ◽  
pp. 423-425 ◽  
Author(s):  
You-Qiang Wang
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Character Degrees ◽  
Finite Solvable Group ◽  
Prime Integer ◽  
Derived Length ◽  
Brauer Character ◽  
Brauer Characters

AbstractLet G be a finite solvable group. Fix a prime integer p and let t be the number of distinct degrees of irreducible Brauer characters of G with respect to the prime p. We obtain the bound 3t — 2 for the derived length of a Hall p'-subgroup of G. Furthermore, if |G| is odd, then the derived length of a Hall p'-subgroup of G is bounded by /.

Character correspondences in solvable groups with a self-normalizing sylow subgroup

2019 ◽  
Vol 19 (10) ◽  
pp. 2050190
Author(s):  
Carolina Vallejo Rodríguez
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Sylow Subgroup ◽  
Solvable Groups ◽  
Character Formula ◽  
Finite Solvable Group ◽  
Irreducible Characters ◽  
Made In

Let [Formula: see text] be a finite solvable group and let [Formula: see text] for some prime [Formula: see text]. Whenever [Formula: see text] is odd, Isaacs described a correspondence between irreducible characters of degree not divisible by [Formula: see text] of [Formula: see text] and [Formula: see text]. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. In the case where [Formula: see text], G. Navarro showed that every irreducible character [Formula: see text] of degree not divisible by [Formula: see text] has a unique linear constituent [Formula: see text] when restricted to [Formula: see text], and that the map [Formula: see text] defines a bijection. Navarro’s bijection is obviously natural in the sense described above. We show that these two correspondences are the same under the intersection of the hypotheses.

scholarly journals Character degrees and derived length in p-groups

1988 ◽  
Vol 30 (2) ◽  
pp. 221-230 ◽  
Author(s):  
Michael C. Slattery
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Character Degrees ◽  
Derived Length ◽  
Odd Order

There are a number of theorems which bound d.l.(G), the derived length of a group G, in terms of the size of the set c.d.(G) of irreducible character degrees of G assuming that G is in some particular class of solvable groups ([1], [3], [4], [7]). For instance, Gluck [4] shows that d.l.(G)≤2 |c.d.(G)| for any solvable group, whereas Berger [1] shows that d.l.(G)≤|c.d.(G)| if G has odd order. One of the oldest (and smallest) such bounds is a theorem of Taketa [7] which says that d.l.(G)≤|c.d.(G)| if G is an M-group. Most of the existing theorems are an attempt to extend Taketa's bound to all solvable groups. However, it is not even known for M-groups whether or not this is the best possible bound. This suggests that given a class of solvable groups one might try to find the maximum derived length of a group with n character degrees (i.e. the best possible bound).

Character Degree Graphs of Solvable Groups of Fitting Height 2

2006 ◽  
Vol 49 (1) ◽  
pp. 127-133 ◽  
Author(s):  
Mark L. Lewis
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Character Degree ◽  
Character Degrees ◽  
Irreducible Characters ◽  
Vertex Set ◽  
Fitting Height ◽  
A Finite Group

AbstractGiven a finite group G, we attach to the character degrees of G a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of G, and with an edge between p and q if pq divides the degree of some irreducible character of G. In this paper, we describe which graphs occur when G is a solvable group of Fitting height 2.

Group Algebras with Minimal Strong Lie Derived Length

2008 ◽  
Vol 51 (2) ◽  
pp. 291-297 ◽  
Author(s):  
Ernesto Spinelli
Keyword(s):  
Solvable Group ◽  
Group Algebra ◽  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Derived Length ◽  
Necessary And Sufficient

AbstractLet KG be a non-commutative strongly Lie solvable group algebra of a group G over a field K of positive characteristic p. In this note we state necessary and sufficient conditions so that the strong Lie derived length of KG assumes its minimal value, namely [log2(p + 1)].

Derived length when a product of characters has two nonprincipal irreducible constituents

2016 ◽  
Vol 15 (06) ◽  
pp. 1650110
Author(s):  
Lisa Rose Hendrixson ◽  
Mark L. Lewis
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Character Formula ◽  
Derived Length

We study the situation where a solvable group [Formula: see text] has a faithful irreducible character [Formula: see text] such that [Formula: see text] has exactly two distinct nonprincipal irreducible constituents. We prove that [Formula: see text] has derived length bounded above by 8, and provide an example of such a group having derived length 8. In particular, this improves upon a result of Adan-Bante.

Numbers Of Conjugacy Class Sizes And Derived Lengths for A-Groups

1996 ◽  
Vol 39 (3) ◽  
pp. 346-351 ◽  
Author(s):  
Mary K. Marshall
Keyword(s):  
Conjugacy Class ◽  
Solvable Group ◽  
Conjugacy Classes ◽  
Finite Solvable Group ◽  
Conjugacy Class Sizes ◽  
Sylow Subgroups ◽  
Derived Length ◽  
Completely Reducible

AbstractAn A-group is a finite solvable group all of whose Sylow subgroups are abelian. In this paper, we are interested in bounding the derived length of an A-group G as a function of the number of distinct sizes of the conjugacy classes of G. Although we do not find a specific bound of this type, we do prove that such a bound exists. We also prove that if G is an A-group with a faithful and completely reducible G-module V, then the derived length of G is bounded by a function of the number of distinct orbit sizes under the action of G on V.

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989 ◽  
Vol 41 (1) ◽  
pp. 68-82 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Group Actions ◽  
Character Degrees ◽  
Prime Divisors ◽  
Irreducible Characters ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
Nonlinear Irreducible Character

The main result of this paper is the following:Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the setcd(G) = ﹛x(l)lx ∈ Irr(G) ﹜of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

LIE SOLVABLE GROUP ALGEBRAS OF DERIVED LENGTH THREE IN CHARACTERISTIC THREE

2012 ◽  
Vol 11 (05) ◽  
pp. 1250098 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Solvable Group ◽  
Commutator Subgroup ◽  
Group Algebras ◽  
Derived Length ◽  
Engel Group

In this paper we provide a characterization of Lie solvable group algebras of derived length three over a field of characteristic three when G is a non-2-Engel group with abelian commutator subgroup.

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