scholarly journals On a Conjecture of G. Malle and G. Navarro on Nilpotent Blocks

10.37236/704 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jean-Baptiste Gramain
Keyword(s):  
Finite Group ◽  
Recent Article ◽  
Simple Groups ◽  
Alternating Group ◽  
Covering Group ◽  
Group 2 ◽  
A Finite Group ◽  
Nilpotent Blocks

In a recent article, G. Malle and G. Navarro conjectured that the $p$-blocks of a finite group all of whose height 0 characters have the same degree are exactly the nilpotent blocks defined by M. Broué and L. Puig. In this paper, we check that this conjecture holds for spin-blocks of the covering group $2.{\mathfrak A}_n$ of the alternating group ${\mathfrak A}_n$, thereby solving a case excluded from the study of quasi-simple groups by Malle and Navarro.

ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

2019 ◽  
Vol 102 (1) ◽  
pp. 77-90
Author(s):  
PABLO SPIGA
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Alternating Group ◽  
Permutation Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

Constructing Representations of Finite Simple Groups and Covers

2006 ◽  
Vol 58 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Vahid Dabbaghian-Abdoly
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Simple Method ◽  
Covering Group ◽  
Representations Of Finite Groups ◽  
A Finite Group

AbstractLet G be a finite group and χ be an irreducible character of G. An efficient and simple method to construct representations of finite groups is applicable whenever G has a subgroup H such that χH has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if G is a simple group or a covering group of a simple group and χ is an irreducible character of G of degree less than 32, then there exists a subgroup H (often a Sylow subgroup) of G such that χH has a linear constituent with multiplicity 1.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

scholarly journals SIMPLE MODULES IN THE AUSLANDER–REITEN QUIVER OF PRINCIPAL BLOCKS WITH ABELIAN DEFECT GROUPS

2018 ◽  
Vol 235 ◽  
pp. 58-85
Author(s):  
SHIGEO KOSHITANI ◽  
CAROLINE LASSUEUR
Keyword(s):  
Finite Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Simple Modules ◽  
Principal Block ◽  
Defect Groups ◽  
Abelian Defect Groups ◽  
A Finite Group

Given an odd prime $p$ , we investigate the position of simple modules in the stable Auslander–Reiten quiver of the principal block of a finite group with noncyclic abelian Sylow $p$ -subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$ , we prove that simple modules in the principal block all lie at the end of their components.

scholarly journals On the Number of p-Regular Elements in Finite Simple Groups

2009 ◽  
Vol 12 ◽  
pp. 82-119 ◽  
Author(s):  
László Babai ◽  
Péter P. Pálfy ◽  
Jan Saxl
Keyword(s):  
Simple Group ◽  
Finite Subgroup ◽  
Simple Groups ◽  
Alternating Group ◽  
Arbitrary Field ◽  
Finite Simple Groups ◽  
Regular Elements ◽  
Exceptional Groups ◽  
Monte Carlo Algorithms ◽  
A Finite Group

AbstractA p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n – 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.Finally we complement our lower bound results with the following upper bound: for all n ≥ 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in PSL(n,q) is less than 3/√n.

scholarly journals Block Extensions, Local Categories and Basic Morita Equivalences

2020 ◽  
Vol 71 (2) ◽  
pp. 703-728
Author(s):  
Tiberiu Coconeţ ◽  
Andrei Marcus ◽  
Constantin-Cosmin Todea
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cohomology Class ◽  
Group Extension ◽  
Modular System ◽  
Pointed Group ◽  
A Finite Group ◽  
Nilpotent Blocks ◽  
The Way

Abstract Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

On mutually permutable products of generalized supersoluble subgroups of finite groups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650054
Author(s):  
E. N. Myslovets
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Paper Properties ◽  
Supersoluble Groups ◽  
A Finite Group ◽  
Mutually Permutable Products ◽  
Composition Factors

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.

scholarly journals A note on centralizers of involutions involving simple groups

1976 ◽  
Vol 14 (3) ◽  
pp. 425-426 ◽  
Author(s):  
Donald Wright
Keyword(s):  
Finite Group ◽  
Simple Groups ◽  
A Finite Group

If a finite group G has a ‘central’ involution t whose centralizer in G is (t) × H then, under certain conditions on H, G cannot be simple.

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