A note on TI and TI defect blocks

by Jianbei An and Charles W. Eaton


Abstract:

We show that every trivial intersection block of a finite group (as introduced 
by Alperin and Brou\'e in \cite{AlperinBroue}) has trivial intersection (TI) 
defect groups, but that the converse is not true in general. We then present 
some conditions equivalent to a block $B$ being a TI block, generalizing the idea of a 
$k$-generated $p$-core to $B$-subgroups. In particular we give further weight 
to Olsson's assertion that TI blocks are a better generalization of groups 
with TI Sylow $p$-subgroups than are TI defect blocks. Finally we describe 
the r\^{o}le of the generalized $k$-generated $p$-core in the control of fusion of subpairs.


Jianbei An: University of Auckland, New Zealand. 
Charles Eaton: University of Birmingham, England. 

Status: preprint (submitted)