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 explicitly construct them for An and Sn for all primes p. Of course, it is known that for p≥5 that An 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.
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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.