On the frobenius group

1992 ◽  
Vol 43 (7-8) ◽  
pp. 954-964
Author(s):  
D. A. Suprunenko
Keyword(s):  
Frobenius Group

Some generalized characters associated to a transitive permutation group

2020 ◽  
Vol 23 (3) ◽  
pp. 393-397
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Transitive Permutation Group ◽  
Point Stabilizer ◽  
Necessary And Sufficient ◽  
Regular Character ◽  
Permutation Character

AbstractLet G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

scholarly journals Frobenius groups of low rank

2021 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Frobenius Group ◽  
Low Rank ◽  
Special Situation ◽  
Frobenius Groups ◽  
Frobenius Kernel

AbstractLet G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

EXTENDING RESULTS OF MORGAN AND PARKER ABOUT COMMUTING GRAPHS

2021 ◽  
pp. 1-9
Author(s):  
NICOLAS F. BEIKE ◽  
RACHEL CARLETON ◽  
DAVID G. COSTANZO ◽  
COLIN HEATH ◽  
MARK L. LEWIS ◽  
...  
Keyword(s):  
Frobenius Group ◽  
Connected Components ◽  
Commuting Graph

Abstract Morgan and Parker proved that if G is a group with ${\textbf{Z}(G)} = 1$ , then the connected components of the commuting graph of G have diameter at most $10$ . Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a $2$ -Frobenius group, and if the commuting graph of G is connected, then its diameter is at most $8$ . We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap {\textbf{Z}(G)} = 1$ . We also prove that if G is solvable and $G/{\textbf{Z}(G)}$ is either a Frobenius group or a $2$ -Frobenius group, then the commuting graph of G is disconnected.

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.

Generalization of the basis property on finite groups

2020 ◽  
pp. 2150111
Author(s):  
Ibrahim Al-Dayel ◽  
Ahmad Al Khalaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Basis Property ◽  
Generating Set ◽  
Equivalent Basis ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Minimal Generating Set

A group [Formula: see text] has the Basis Property if every subgroup [Formula: see text] of [Formula: see text] has an equivalent basis (minimal generating set). We studied a special case of the finite group with the Basis Property, when [Formula: see text]-group [Formula: see text] is an abelian group. We found the necessary and sufficient conditions on an abelian [Formula: see text]-group [Formula: see text] of [Formula: see text] with the Basis Property to be kernel of Frobenius group.

Finite Groups with Four Relative Commutativity Degrees

2015 ◽  
Vol 22 (03) ◽  
pp. 449-458 ◽  
Author(s):  
A. Erfanian ◽  
M. Farrokhi D.G.
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Maximal Subgroups ◽  
Frobenius Kernel ◽  
A Finite Group

It is shown that a finite group G has four relative commutativity degrees if and only if G/Z(G) is a p-group of order p3 and G has no abelian maximal subgroups, or G/Z(G) is a Frobenius group with Frobenius kernel and complement isomorphic to ℤp × ℤp and ℤq, respectively, and the Sylow p-subgroup of G is abelian, where p and q are distinct primes.

scholarly journals Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

2014 ◽  
Vol 42 (11) ◽  
pp. 4751-4756 ◽  
Author(s):  
Gülin Ercan ◽  
İsmail Ş. Güloğlu ◽  
Elif Öğüt
Keyword(s):  
Solvable Group ◽  
Frobenius Group ◽  
Finite Solvable Group ◽  
Group Of Automorphisms ◽  
Nilpotent Length
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