Nonabelian Defect Groups Research Articles

Isotypies for p-blocks with non-abelian metacyclic defect groups, p odd

Let p be an odd prime, G be a finite group and b be a p-block of G with non-abelian metacyclic defect group P. Then it is known that a hyperfocal subgroup Q of b is cyclic. In this study motivated by Rouquier's conjecture on blocks with abelian hyperfocal subgroups, we show that b is isotypic to its Brauer correspondents in NG(P) and NG(Q).

On the complexities of some simple modules of symmetric groups

Let p be a prime. In this paper, we compute the complexities of some simple modules of symmetric groups labelled by two-part partitions. Most of the simple modules considered here are contained in p-blocks with non-abelian defect groups.

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.

Loewy lengths of centers of blocks

Let B be a block of a finite group with respect to an algebraically closed field F of characteristic p>0. In a recent paper, Otokita gave an upper bound for the Loewy length LL(ZB) of the center ZB of B in terms of a defect group D of B. We refine his methods in order to prove the optimal bound LL(ZB)\le LL(FD) whenever D is abelian. We also improve Otokita's bound for non-abelian defect groups. As an application we classify the blocks B such that LL(ZB)\ge |D|/2.

Cartan matrices and Brauer's $k(B)$-conjecture IV

In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a $p$-block of a finite group with abelian defect group $D$ is bounded by $|D|$ (Brauer's $k(B)$-conjecture) provided $D$ has no large elementary abelian direct summands. Moreover, we verify Brauer's $k(B)$-conjecture for all blocks with minimal non-abelian defect groups. This extends previous results by various authors.

On the number of simple modules in a block of a finite group

We prove that if B is a p-block with non-trivial defect group D of a finite p-solvable group G, then ℓ(B)<pr, where r is the sectional rank of D. We remark that there are infinitely many p-blocks B with non-Abelian defect groups and ℓ(B)=pr−1.We conjecture that the inequality ℓ(B)≤pr holds for an arbitrary p-block with defect group of sectional rank r. We show this to hold for a large class of p-blocks of various families of quasi-simple and nearly simple groups.

2-Blocks with minimal nonabelian defect groups III

2-Blocks with minimal nonabelian defect groups III

2-Blocks with minimal nonabelian defect groups III

2-Blocks with minimal nonabelian defect groups III

The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups

We prove the Alperin-McKay Conjecture for all p-blocks of finite groups with metacyclic, minimal nonabelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order p. In the special case p = 3, we also verify Alperin’s Weight Conjecture for these defect groups. Moreover, in case p = 5 we do the same for the non-abelian defect groups C25 o C5n . The proofs do not rely on the classification of the finite simple groups.

Characters of Positive Height in Blocks of Finite Quasi-Simple Groups

Eaton and Moreto proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~$p$. We provide some further evidence for blocks of groups of Lie type considered in cross characteristic.

Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups

We propose a generalization of Brauer?s Height Zero Conjecture that considers positive heights. We give strong evidence supporting one half of the generalization and obtain some partial results regarding the other half.

A note on Olssonʼs Conjecture

For a p-block B of a finite group G with defect group D Olsson conjectured that k0(B)⩽|D:D′|, where k0(B) is the number of characters in B of height 0 and D′ denotes the commutator subgroup of D. Brauer deduced Olssonʼs Conjecture in the case where D is a dihedral 2-group using the fact that certain algebraically conjugate subsections are also conjugate in G. We generalize Brauerʼs argument for arbitrary primes p and arbitrary defect groups. This extends two results by Robinson. For p>3 we show that Olssonʼs Conjecture is satisfied for defect groups of p-rank 2 and for minimal non-abelian defect groups.

2-Blocks with minimal nonabelian defect groups II

Abstract. We determine the structure of 2-blocks with minimal nonabelian defect groups, by making use of the classification of finite simple groups.

Blocks of full defect with nonabelian metacyclic defect groups

Let p be an odd prime and let B be a p-block of a finite group G with a nonabelian metacyclic defect group P which is a Sylow p-subgroup of G. The purpose of this article is to study the ordinary and modular irreducible characters in B. In particular, we calculate ki(B) and li(B) for an arbitrary nonnegative integer i.

2-Blocks with minimal nonabelian defect groups

We study numerical invariants of 2-blocks with minimal nonabelian defect groups. These groups were classified by Rédei (see Rédei, 1947 [41]). If the defect group is also metacyclic, then the block invariants are known (see Sambale [43]). In the remaining cases there are only two (infinite) families of ‘interesting’ defect groups. In all other cases the blocks are nilpotent. We prove Brauerʼs k ( B ) -conjecture and Olssonʼs conjecture for all 2-blocks with minimal nonabelian defect groups. For one of the two families we also show that Alperinʼs weight conjecture and Dadeʼs conjecture are satisfied. This paper is a part of the authorʼs PhD thesis.

Blocks with nonabelian defect groups which have cyclic subgroups of index p

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and l(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and l(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group Mn+1(p) of order pn+1 and exponent pn for \({n \geqslant 2}\), and where A is not necessarily a full defect p-block but its defect group P = Mn+1(p) is normal in G. The proof is independent of the classification of finite simple groups.

On Brauer's [formula omitted]-problem for blocks of p-solvable groups with non-Abelian defect groups

On Brauer's [formula omitted]-problem for blocks of p-solvable groups with non-Abelian defect groups