Defect zero p-blocks for finite simple groups

@article{Granville1996DefectZP,
  title={Defect zero p-blocks for finite simple groups},
  author={Andrew Granville and Ken Ono},
  journal={Transactions of the American Mathematical Society},
  year={1996},
  volume={348},
  pages={331-347}
}
  • A. Granville, K. Ono
  • Published 1 December 1996
  • Mathematics
  • Transactions of the American Mathematical Society
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p-blocks remained unclassified were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture… 

Heights of characters and defect groups

An important result in ordinary character theory is the Ito-Michler theorem, which asserts that a prime p does not divide the degree of any irreducible character of a finite group G if and only if G

Blocks of small defect in alternating groups and squares of Brauer character degrees

Abstract Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G

Blocks of small defect in symmetric and simple groups

ABSTRACT Let p be a prime, G a finite group, and r≥1. We say a p-block B with defect group D of G is r-small if . In this paper, we show that almost all simple groups have r-small p-blocks, and we

On the asymptotics of the number of p̅-core partitions of integers

0. Introduction. In the following the symbol p always denotes an odd prime number and n a natural number. In the representation theory of the symmetric groups Sn there is some interest in the

4-core partitions and class numbers

Let Ct(n) denote the number of t−core partitions of n, where a partition is a t−core if none of the hook numbers of the associated Ferrers-Young diagram are multiples of t. It is well known that

Zeros of Brauer characters and the defect zero graph

Abstract We show that for each finite non-abelian simple group and each prime p either there exists an irreducible Brauer character which takes the value zero on some p-regular element, or p = 2 and

On the existence of rook equivalent t-cores

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

In this note, we show that if a finite group G has only one conjugacy class of subgroups of an odd prime order p, then G has a p-block of defect zero if and only if Op(G) = 1. Generally, this result
...

References

SHOWING 1-10 OF 23 REFERENCES

Landau's theorem for p-blocks of p-solvable groups.

Landau's theorem (see [10]) says that, for any positive integer fc, there are only fmitely many isomorphism classes of finite groups with exactly k conjugacy classes. Thus, for any positive integer

Finite Simple Groups: An Introduction to Their Classification

In February 1981, the classification of the finite simple groups (Dl)* was completed,t. * representing one of the most remarkable achievements in the history or mathematics. Involving the combined

A Note on the Number of $t$-Core Partitions

A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a partition in the natural way. Fix a positive integer t. A partition

Some congruences for partitions that are p-cores

A number of linear congruences modulo r are proved for the number of partitions that are p-cores where p is prime, 5≤p≤23, and r is any prime divisor of 1/2(p-1). Analogous results are derived for

Modular forms and representations of symmetric groups

We give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups. Using this we can obtain asymptotic formulas for the number of blocks of

CRANKS AND T -CORES

New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the

The theory of partitions

1. The elementary theory of partitions 2. Infinite series generating functions 3. Restricted partitions and permutations 4. Compositions and Simon Newcomb's problem 5. The Hardy-Ramanujan-Rademacher

Introduction to Elliptic Curves and Modular Forms

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book

Blocks for symmetric groups and their covering groups and quadratic forms.

Introduction. Let p be a prime number, K a eld of characteristic p and let S n be the group of permu