scholarly journals The dominant dimension of cohomological Mackey functors

2018 ◽  
Vol 17 (12) ◽  
pp. 1850228
Author(s):  
Markus Linckelmann
Keyword(s):  
Finite Group ◽  
Defect Group ◽  
Dominant Dimension ◽  
Text Block ◽  
Symmetric Algebras ◽  
Endomorphism Algebras ◽  
Mackey Functors ◽  
A Finite Group

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category [Formula: see text] of cohomological Mackey functors of a [Formula: see text]-block [Formula: see text] of a finite group with a nontrivial defect group is [Formula: see text].

scholarly journals ON PERFECT ISOMETRIES FOR TAME BLOCKS

2002 ◽  
Vol 34 (1) ◽  
pp. 46-54 ◽  
Author(s):  
RADHA KESSAR ◽  
MARKUS LINCKELMANN
Keyword(s):  
Finite Group ◽  
Defect Group ◽  
Morita Equivalent ◽  
A Finite Group

Any 2-block of a finite group G with a quaternion defect group Q8 is Morita equivalent to the corresponding block of the centraliser H of the unique involution of Q8 in G; this answers positively an earlier question raised by M. Broué.

scholarly journals A Note on a Conjecture of Brauer

1963 ◽  
Vol 22 ◽  
pp. 1-13 ◽  
Author(s):  
Paul Fong
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Solvable Group ◽  
Prime Divisor ◽  
Irreducible Character ◽  
Defect Group ◽  
Irreducible Characters ◽  
A Finite Group

In [1] R. Brauer asked the following question: Let be a finite group, p a rational prime number, and B a p-block of with defect d and defect group . Is it true that is abelian if and only if every irreducible character in B has height 0 ? The present results on this problem are quite incomplete. If d-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that if is cyclic, then no characters of positive height appear in B. If is normal in , the conjecture was proved by W. Reynolds and M. Suzuki, [12]. In this paper we shall show that for a solvable group , the conjecture is true for the largest prime divisor p of the order of . Actually, one half of this has already been proved in [7]. There it was shown that if is a p-solvable group, where p is any prime, and if is abelian, then the condition on the irreducible characters in B is satisfied.

Robinson’s conjecture on heights of characters

2019 ◽  
Vol 155 (6) ◽  
pp. 1098-1117 ◽  
Author(s):  
Zhicheng Feng ◽  
Conghui Li ◽  
Yanjun Liu ◽  
Gunter Malle ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Defect Group ◽  
Character Degrees ◽  
Simple Groups ◽  
Reductive Groups ◽  
A Finite Group

Geoffrey Robinson conjectured in 1996 that the $p$-part of character degrees in a $p$-block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes $p\neq 2$ for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then studied using deep results on blocks of finite reductive groups.

Blocks of defect zero in finite groups with conjugate subgroups of a given prime order

2017 ◽  
Vol 16 (11) ◽  
pp. 1750217
Author(s):  
Tianze Li ◽  
Yanjun Liu ◽  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Groups ◽  
Prime Order ◽  
Text Block ◽  
Order Group ◽  
Odd Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] be a prime. In this note, we show that if [Formula: see text] and all subgroups of [Formula: see text] of order [Formula: see text] are conjugate, then either [Formula: see text] has a [Formula: see text]-block of defect zero, or [Formula: see text] and [Formula: see text] is a direct product of a simple group [Formula: see text] and an odd order group. This improves one of our previous works.

scholarly journals Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

2015 ◽  
Vol 25 (06) ◽  
pp. 951-976 ◽  
Author(s):  
Benjamin Sambale
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Automorphism Groups ◽  
Defect Group ◽  
Sporadic Simple Group ◽  
Sporadic Groups ◽  
A Finite Group ◽  
Group D

Let B be a p-block of a finite group G with abelian defect group D such that S ≤ G ≤ Aut (S), S′ = S and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N G(D) in the sense of Broué. This has been done by Rouquier for principal blocks and it remains to deal with the non-principal blocks.

scholarly journals Group graded basic Morita equivalences and the Harris–Knörr correspondence

2020 ◽  
Vol 23 (4) ◽  
pp. 697-708
Author(s):  
Andrei Marcus
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Morita Equivalence ◽  
Defect Group ◽  
A Finite Group

AbstractLet G be a finite group, let b be a G-invariant block with defect group Q of the normal subgroup H of G, and let {b^{\prime}\in Z(\mathcal{O}N_{H}(Q))} be the Brauer correspondent of b. We show that the bijection between the blocks of G covering b and the blocks of {N_{G}(Q)} covering {b^{\prime}}, induced by a {G/H}-graded basic Morita equivalence between the block extensions {b\mathcal{O}G} and {b^{\prime}\mathcal{O}N_{G}(Q)}, coincides with the Harris–Knörr correspondence.

scholarly journals Walking around the Brauer tree

1974 ◽  
Vol 17 (2) ◽  
pp. 197-213 ◽  
Author(s):  
J. A. Green
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Defect Group ◽  
Finite Characteristic ◽  
Decomposition Matrix ◽  
Linear Graph ◽  
Famous Paper ◽  
A Finite Group

LetG be a finite group, and k a field of finite characteristic p, such that the polynomial x¦G¦ –1 splits completely in k[x]. Let Β be a kG-block which has defect group D which is cylclic of order pd (d ≧ 1). Brauer showed in a famous paper [2] that, in case d = 1, the decomposition matrix of Β is determined by a certain positive integer e which divides p − 1, and a tree Г, a connected acyclic linear graph of e + 1 vertices and e edges. Twenty-five years later Dade ([3]) extended Brauer's theorem to the general case.

The Existence of p-Blocks of a Finite Group

2006 ◽  
Vol 13 (04) ◽  
pp. 705-710
Author(s):  
Fangsheng Qian
Keyword(s):  
Finite Group ◽  
Regular Element ◽  
Defect Group ◽  
P Elements ◽  
A Finite Group ◽  
Mod P

Suppose G is a finite group and D is a normal p-subgroup of G with |D|=pd. Let GP denote the set of p-elements of G and Φ(g)={(a,b) ∈ GP× GP | ab=g}. We show that G has a p-block with D as a defect group if and only if there exists a p-regular element g of G with D as a Sylow p-subgroup of CG(g) such that |Φ(g)| ≠ 0 (mod pd+1).

scholarly journals Robinson’s conjecture for classical groups

2019 ◽  
Vol 22 (4) ◽  
pp. 555-578 ◽  
Author(s):  
Zhicheng Feng ◽  
Conghui Li ◽  
Yanjun Liu ◽  
Gunter Malle ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Defect Group ◽  
Simple Groups ◽  
Classical Groups ◽  
Central Quotient ◽  
Ordinary Character ◽  
A Finite Group

AbstractRobinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks of finite quasi-simple classical groups.

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