# On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras

@article{Koshitani2019OnTO, title={On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras}, author={Shigeo Koshitani and Taro Sakurai}, journal={Archiv der Mathematik}, year={2019}, volume={113}, pages={1-10} }

In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block B with $$k(B) = 1$$k(B)=1 where k(B) is the number of irreducible ordinary characters of B. In 1982, Brandt established a characterization of a block with defect group of order two as a block B with $$k(B) = 2$$k(B)=2. These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively. In this paper… Expand

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#### References

SHOWING 1-10 OF 22 REFERENCES

Symmetric local algebras and small blocks of finite groups

- Mathematics
- 1984

One of the major problems in modular representation theory is the deter- mination of the numbers k(B) of ordinary and Z(B) of modular irreducible characters for the block B of a finite group G with… Expand

On blocks of defect two and one simple module, and Lie algebra structure of $HH^1$

- Mathematics
- 2016

Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over $k$. As a consequence,… Expand

Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

- Mathematics
- 2015

Abstract We provide a description of the torsion subgroup 𝑇𝑇 ( G ) ${\mathit{TT}(G)}$ of the finitely generated abelian group T ( G ) ${T(G)}$ of endo-trivial kG-modules in the case that G has… Expand

Rings and Categories of Modules

- Mathematics
- 1974

This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring… Expand

Finite group algebras and their modules

- Mathematics
- 1983

Preface Part I. The Structure of Group Algebras: 1. Idempotents in rings. Liftings 2. Projective and injective modules 3. The radical and artinian rings 4. Cartan invariants and blocks 5. Finite… Expand

Central elements of the Jennings basis and certain Morita invariants

- Mathematics
- Journal of Algebra and Its Applications
- 2019

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the [Formula: see text]th (right) socle [Formula: see text] of a… Expand

On Frobenius algebras

- Mathematics
- 2003

Abstract All Frobenius algebras Γ with separable factor algebra Γ /rad Γ are constructed as factors of tensor algebras. Further, fixing a separable algebra A , a bimodule A V A and a natural number… Expand

Symmetric local algebras with 5-dimensional center

- Mathematics
- 1992

We prove that a symmetric split local algebra whose center is 5-dimensional has dimension 5 or 8. This implies that the defect groups of a block of a finite group containing exactly fivbe irreducible… Expand

On diagonal entries of Cartan matrices of p-blocks

- Mathematics
- 2016

In this short note, we show some inequalities on Cartan matrices, centers and socles of blocks of group algebras. Our main theorems are generalizations of the facts on dimension of Reynolds ideals.

Some inequalities between invariants of blocks

- Mathematics
- 2002

Abstract. We prove inequalities between the number of irreducible ordinary characters in a p-block B of a finite group G and Cartan invariants of B, using integral quadratic forms.