Permutation Modules, Graphs, and Blocks

Abstract

In this paper we give another method to determine the existence of blocks with given defect group. At the end of this paper, a new interpretation of a result in Iizuka and Watanabe (Iizuka, K., Watanabe, A. (1973). On the number of blocks of irreducible characters of a finite group with a given defect group. Kumamoto. J. Sci. (Math.) 9: 55–61) is given.

Key Words:

• Permutation module
• Block
• Digraph
• Vertex
• Defect group
• Double coset

2000 Mathematics Subject Classification:

• Primary 20C20

Acknowledgments

Lizhong Wang was Supported by NNSFC and RFDP. The author is grateful to Professor Zhang and Professor Shi for their encouragement and useful suggestions. The author is also grateful to the referee for his or her kindness and crucial suggestions for revisions.