Search results for "Finite group"

Weights and Nilpotent Subgroups

2018

In a finite group G, we consider nilpotent weights, and prove a pi-version of the Alperin Weight Conjecture for certain pi-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the first author.

Near abelian profinite groups

2012

Abstract A compact p-group G (p prime) is called near abelian if it contains an abelian normal subgroup A such that G/A has a dense cyclic subgroup and that every closed subgroup of A is normal in G. We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called quasihamiltonian (or modular) if every pair of compact subgroups commutes setwise. We show that for p ≠ 2 a compact p-group G is near abelian if and only if it is quasihamiltonian. The case p = 2 is discussed separately.

Representations of Finite Groups

2009

Uncountable existentially closed groups in locally finite group classes

1990

In this paper, will always denote a local class of locally finite groups, which is closed with respect to subgroups, homomorphic images, extensions, and with respect to cartesian powers of finite -groups. Examples for x are the classes L ℐπ of all locally finite π-groups and L(ℐπ ∩ ) of all locally soluble π-groups (where π is a fixed set of primes). In [4], a wreath product construction was used in the study of existentially closed -groups (=e.c. -groups); the restrictive type of construction available in [4] permitted results for only countable groups. This drawback was then removed partially in [5] with the help of permutational products. Nevertheless, the techniques essentially only per…

Homogeneous actions on the random graph

2018

We show that any free product of two countable groups, one of them being infinite, admits a faithful and homogeneous action on the Random Graph. We also show that a large class of HNN extensions or free products, amalgamated over a finite group, admit such an action and we extend our results to groups acting on trees. Finally, we show the ubiquity of finitely generated free dense subgroups of the automorphism group of the Random Graph whose action on it have all orbits infinite.

Fields of values of odd-degree irreducible characters

2019

Abstract In this paper we clarify the quadratic irrationalities that can be admitted by an odd-degree complex irreducible character χ of an arbitrary finite group. Write Q ( χ ) to denote the field generated over the rational numbers by the values of χ, and let d > 1 be a square-free integer. We prove that if Q ( χ ) = Q ( d ) then d ≡ 1 (mod 4) and if Q ( χ ) = Q ( − d ) , then d ≡ 3 (mod 4). This follows from the main result of this paper: either i ∈ Q ( χ ) or Q ( χ ) ⊆ Q ( exp ⁡ ( 2 π i / m ) ) for some odd integer m ≥ 1 .

The set of character degrees of a finite group does not determine its solvability

2014

Degrees of characters in the principal block

2021

Abstract Let G be a finite group. We prove that if the set of degrees of characters in the principal p-block of G has size at most 2 then G is p-solvable, and G / O p ′ ( G ) has a metabelian normal Sylow p-subgroup. The general question of proving that if an arbitrary p-block has two degrees then their defect groups are metabelian remains open.

The set of conjugacy class sizes of a finite group does not determine its solvability

2014

Abstract We find a pair of groups, one solvable and the other non-solvable, with the same set of conjugacy class sizes.

Maximal subgroups and formations

1989

Abstract We define, in each finite group G , some subgroups of Frattini-type in relation with a saturated formation and with a set of primes and study their properties, especially their influence in the structure of G .