Equivalence of normality and characteristicity conditions for conjugacy functor
This article gives a proof/explanation of the equivalence of multiple definitions for the term conjugacy functor that gives a normal subgroup
View a complete list of pages giving proofs of equivalence of definitions
This article gives a proof/explanation of the equivalence of multiple definitions for the term characteristic p-functor that gives a characteristic subgroup
View a complete list of pages giving proofs of equivalence of definitions
Contents
Statement
For conjugacy functors arising from characteristic p-functors
Suppose is a prime number and
is a finite group such that
is a conjugacy functor for
for the prime
arising from a characteristic p-functor. The following are equivalent:
- For every pair of
-Sylow subgroups
of
,
.
- For every pair of
-Sylow subgroups
of
,
is a normal subgroup of
.
- Each of these:
-
is a weakly closed conjugacy functor and there exists a
-Sylow subgroup
of
such that
where
is the p-core of
.
-
is a weakly closed conjugacy functor and for every
-Sylow subgroup
of
,
where
is the
-core of
.
-
- Each of these:
- There exists a
-Sylow subgroup
of
such that
is a characteristic subgroup of
.
- For every
-Sylow subgroup
of
,
is a characteristic subgroup of
.
- There exists a
- Each of these:
- There exists a
-Sylow subgroup
of
such that
is a normal subgroup of
.
- For every
-Sylow subgroup
of
,
is a normal subgroup of
.
- There exists a
For conjugacy functors not arising from characteristic p-functors
All parts of the statement above are equivalent with the exception of (4).
Related facts
Facts used
- Sylow implies order-conjugate
- Equivalence of definitions of weakly closed conjugacy functor
- Characteristic implies normal
- Normality satisfies intermediate subgroup condition
Proof
Characteristic p-functor case
The proofs of the equivalence of both versions of (3) with each other, both versions of (4) with each other, and both versions of (5) with each other follow from Fact (1).
The equivalence of (1), (2), and (3) essentially follows from Fact (2).
(1) implies (4) is easy to see if it is a characteristic p-functor: we have given a definition: "pick a Sylow subgroup and apply to it" that always yields the same subgroup. This is a subgroup-defining function and hence gives a characteristic subgroup of
.
(4) implies (5) follows from Fact (3).
Finally, we show (5) implies (2): then on account of being a normal
-subgroup. Hence,
for all
, and thus by Fact (4), it is normal in
.
General case
In the general case, the proof proceeds the same way, but instead of showing (1) implies (4) and (4) implies (5), we directly show that (1) implies (5).