# Broué's abelian defect group conjecture holds for the Harada-Norton sporadic simple group $HN$ - Mathematics > Representation Theory

Broué's abelian defect group conjecture holds for the Harada-Norton sporadic simple group $HN$ - Mathematics > Representation Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: In representation theory of finite groups, there is a well-known andimportant conjecture due to M. Brou\-e. He conjectures that, for any prime $p$,if a $p$-block $A$ of a finite group $G$ has an abelian defect group $P$, then$A$ and its Brauer corresponding block $B$ of the normaliser $N GP$ of $P$ in$G$ are derived equivalent Rickard equivalent. This conjecture is calledBrou\-e-s abelian defect group conjecture. We prove in this paper thatBrou\-e-s abelian defect group conjecture is true for a non-principal 3-block$A$ with an elementary abelian defect group $P$ of order 9 of the Harada-Nortonsimple group $HN$. It then turns out that Brou\-e-s abelian defect groupconjecture holds for all primes $p$ and for all $p$-blocks of the Harada-Nortonsimple group $HN$.

Author: ** Shigeo Koshitani, Jürgen Müller**

Source: https://arxiv.org/