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 $\left|D\right|\le \sqrt[r]{|G{|}_p}$. In this paper, we show that almost all simple groups have r-small p-blocks, and we explicitly construct them for A_n and S_n for all primes p. Of course, it is known that for p≥5 that A_n has a block of defect zero, but little is known about specifically constructing such blocks. Here, however, our explicit construction of r-small p-blocks allows us to construct certain oddly behaved embeddings of q-blocks into p-blocks, where q is a prime distinct from p.

KEYWORDS:

• Alternating groups
• defect groups
• simple groups
• symmetric groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 20C20
• Secondary 20C15
• 20B15

Acknowledgements

The author would like to thank Hung Tong-Viet for his invaluable assistance with the content of the final section. In addition, the referee provided numerous helpful comments and corrections, and the author would like to thank him or her.