On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

Abstract

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.

Key Words:

• Automorphism groups
• Centralizers
• Coxeter groups
• Indecomposability
• Isomorphism problem

AMS Subject Classification (2000):

• 20F55
• 20E34
• 20F28

ACKNOWLEDGMENTS

The author is supported by JSPS Research Fellowship (No. 16-10825).

I would like to express my deep gratitude to everyone who helped me, especially to Professors Itaru Terada and Kazuhiko Koike for their precious advice and encouragement, and also to the referee for the careful examination and several important suggestions for improvement.

Notes

Communicated by A. Olshanskii.

Additional information

Notes on contributors

Koji Nuida

*Current affiliation: Research Center for Information Security (RCIS), National Institute of Advanced Industrial Science and Technology (AIST), Akihabara-Daibiru Room 1102, 1-18-13 Sotokanda, Chiyoda-ku, Tokyo 101-0021, Japan; Fax: [phone]; [email protected]