On Loewy lengths of blocks

2014 ◽  
Vol 156 (3) ◽  
pp. 555-570 ◽  
Author(s):  
SHIGEO KOSHITANI ◽  
BURKHARD KÜLSHAMMER ◽  
BENJAMIN SAMBALE
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Solvable Groups ◽  
Simple Groups ◽  
Short List ◽  
Loewy Length ◽  
Defect Groups ◽  
Cyclic Defect Group ◽  
A Finite Group ◽  
Groups Of Lie Type

AbstractWe give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p=2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

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.

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.

A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

2005 ◽  
Vol 12 (03) ◽  
pp. 431-442 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Natural Numbers ◽  
Prime Graphs ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pkand natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Apwith p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

A Class of Finite Groups with Abelian Sylow p-Subgroups

2006 ◽  
Vol 13 (03) ◽  
pp. 471-480
Author(s):  
Zhikai Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Defect Groups ◽  
Abelian Defect Groups ◽  
A Finite Group

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

Mersenne primes and solvable Sylow numbers

2016 ◽  
Vol 15 (09) ◽  
pp. 1650163
Author(s):  
Tian-Ze Li ◽  
Yan-Jun Liu
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Solvable Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Nonabelian Simple Group ◽  
Mersenne Primes ◽  
A Finite Group ◽  
Mersenne Prime

Let [Formula: see text] be a prime. The Sylow [Formula: see text]-number of a finite group [Formula: see text], which is the number of Sylow [Formula: see text]-subgroups of [Formula: see text], is called solvable if its [Formula: see text]-part is congruent to [Formula: see text] modulo [Formula: see text] for any prime [Formula: see text]. P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow [Formula: see text]-number of a finite group is the product of two kinds of factors: of prime powers [Formula: see text] with [Formula: see text] (mod [Formula: see text]) and of the number of Sylow [Formula: see text]-subgroups in certain finite simple groups (involved in [Formula: see text]). These classical results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, we show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to [Formula: see text] for [Formula: see text] a Mersenne prime.

scholarly journals Elements with Square Roots in Finite Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 677-690 ◽  
Author(s):  
M. S. Lucido ◽  
M. R. Pournaki
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Square Root ◽  
Alternating Groups ◽  
Square Roots ◽  
A Finite Group ◽  
Groups Of Lie Type

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

Recognizing Finite Groups Through Order and Degree Pattern

2008 ◽  
Vol 15 (03) ◽  
pp. 449-456 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Simple Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups, alternating groups Ap (p ≥ 5 is a twin prime) and some simple groups of Lie type. In this paper, we continue this investigation. In particular, we show that the automorphism groups of sporadic simple groups (except Aut (J2) and Aut (McL)), all simple C22-groups, the alternating groups Ap, Ap+1, Ap+2 and the symmetric groups Sp, Sp+1, where p is a prime, are also uniquely determined by their degree patterns and orders.

scholarly journals The Projective Cover of the Trivial Representation for a Finite Group of Lie Type in Defining Characteristic

2017 ◽  
Vol 24 (03) ◽  
pp. 439-452
Author(s):  
Shigeo Koshitani ◽  
Jürgen Müller
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Finite Field ◽  
Projective Cover ◽  
Arbitrary Field ◽  
Trivial Representation ◽  
Characteristic P ◽  
Loewy Length ◽  
Trivial Module ◽  
A Finite Group

We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra kG of a finite group G of Lie type defined over a finite field of odd characteristic p, where k is an arbitrary field of characteristic p. The proof uses Auslander-Reiten theory.

scholarly journals Element orders in covers of finite simple groups of Lie type

2015 ◽  
Vol 14 (04) ◽  
pp. 1550056 ◽  
Author(s):  
Mariya A. Grechkoseeva
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Element Orders ◽  
A Finite Group ◽  
Groups Of Lie Type

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. We study element orders in proper covers of a finite simple group L of Lie type and prove that such a cover always contains an element whose order differs from the element orders of L provided that L is not L4(q), U3(q), U4(q), U5(2), or 3D4(2).

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

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