Groups satisfying the condition of weak minimality for subgroups of derived length two

1994 ◽  
Vol 46 (9) ◽  
pp. 1322-1326
Author(s):  
V. A. Onishchuk ◽  
Ya. P. Sysak
Keyword(s):  
Derived Length ◽  
Weak Minimality

COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350044
Author(s):  
TIBOR JUHÁSZ ◽  
ENIKŐ TÓTH
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Group Algebras ◽  
Augmentation Ideal ◽  
Characteristic P ◽  
Derived Length ◽  
Commutator Identity ◽  
Nilpotency Index ◽  
Powerful Group

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

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)].

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].

scholarly journals A group of exponent $4$ with derived length at least $4$

1973 ◽  
Vol 39 (2) ◽  
pp. 228-228
Author(s):  
S. Bachmuth ◽  
H. Y. Mochizuki ◽  
K. Weston
Keyword(s):  
Derived Length

scholarly journals The Derived Length of p-Groups

2000 ◽  
Vol 224 (2) ◽  
pp. 263-267 ◽  
Author(s):  
Avinoam Mann
Keyword(s):  
Derived Length

scholarly journals Triangles of Baumslag–Solitar Groups

2012 ◽  
Vol 64 (2) ◽  
pp. 241-253 ◽  
Author(s):  
Daniel Allcock
Keyword(s):  
Finite Groups ◽  
Derived Length ◽  
Special Cases ◽  
Finite Quotients ◽  
General Conditions

Abstract Our main result is that many triangles of Baumslag–Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag–Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length ≤ 3. We obtain a lot of information about finite quotients, even when we cannot determine developability.

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.

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 /.

scholarly journals Some challenging group presentations

1999 ◽  
Vol 67 (2) ◽  
pp. 206-213 ◽  
Author(s):  
George Havas ◽  
Derek F. Holt ◽  
P. E. Kenne ◽  
Sarah Rees
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Derived Length ◽  
Finitely Presented Group ◽  
Group Presentations ◽  
Coset Enumeration ◽  
Finitely Presented

AbstractWe study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

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