Abstract
Abstract
For a finite abelian p-group A and a subgroup
$\Gamma \le \operatorname {\mathrm {Aut}}(A)$
, we say that the pair
$(\Gamma ,A)$
is fusion realizable if there is a saturated fusion system
${\mathcal {F}}$
over a finite p-group
$S\ge A$
such that
$C_S(A)=A$
,
$\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $
as subgroups of
$\operatorname {\mathrm {Aut}}(A)$
, and
. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for
$p=2$
or
$3$
and
$\Gamma $
one of the Mathieu groups, that the only
${\mathbb {F}}_p\Gamma $
-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.
Publisher
Cambridge University Press (CUP)