Weakly closed subgroup for a fusion system
From Groupprops
This article defines a property that can be evaluated for a group of prime power order, equipped with a fusion system
View other such properties
Contents
Definition
Suppose is a group of prime power order and
is a fusion system on
. Then a subgroup
is termed weakly closed with respect to
if for every morphism
in
, we have
.
Relation with other properties
Stronger properties
Related subgroup properties and subgroup-of-subgroup properties
- Weakly closed subgroup: Suppose
are groups.
is weakly closed in
with respect to
if, for any
,
implies that
.
- Weakly closed subgroup of Sylow subgroup: The case where
is a
-Sylow subgroup of
. This is related to fusion systems as follows: a subgroup of a
-Sylow subgroup
of a finite group
is weakly closed in
if and only if it is weakly closed for the fusion system induced by
on
.
- Sylow-relatively weakly closed subgroup: A subgroup of a group of prime power order that is a weakly closed subgroup in any group containing the bigger group as a Sylow subgroup.
- Fusion system-relatively weakly closed subgroup: A subgroup of a group of prime power order that is weakly closed in the fusion system sense).for any fusion system on the bigger group.