Oliver subgroup
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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Contents
Definition
The main Oliver subgroup
Suppose is a prime number and
is a finite p-group. The Oliver subgroup of
, denoted
(don't know what that letter actually is) is defined as the unique largest subgroup of
such that there exists an ascending series of subgroups of
:
such that:
- Each
is a normal subgroup of
.
- We have the condition that for all
where the indicates an iterated commutator of the form
with
occurring
times.
Here, denotes the first omega subgroup.
Other Oliver subgroups
Suppose is a prime number and
is a finite p-group. The
Oliver subgroup of
, denoted
(don't know what that letter actually is) is defined as the unique largest subgroup of
such that there exists an ascending series of subgroups of
:
such that:
- Each
is a normal subgroup of
.
- We have the condition that for all
where the indicates an iterated commutator of the form
with
occurring
times.
Here, denotes the first omega subgroup.
Relation between definitions
The main Oliver subgroup is the Oliver subgroup.