Normal Hall implies permutably complemented
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal Hall subgroup) must also satisfy the second subgroup property (i.e., permutably complemented subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about normal Hall subgroup|Get more facts about permutably complemented subgroup
Name
This result is sometimes called Schur's theorem, and is considered a part of the Schur-Zassenhaus theorem, which also asserts that any two permutable complements to a normal Hall subgroup are conjugate. The second half is available at Hall retract implies order-conjugate.
Statement
Verbal statement
Any normal Hall subgroup of a group is permutably complemented.
Statement with symbols
Suppose is a normal Hall subgroup of a group
. Then there exists a subgroup
such that
is trivial and
. Note that this makes
a Hall subgroup as well.
Equivalently, if is a normal
-Hall subgroup of
, then
possesses a
-Hall subgroup: a Hall subgroup corresponding to the primes not in
.
Facts used
- Abelian normal Hall implies permutably complemented: This was the original version of the theorem proved by Schur.
- Sylow subgroups exist
- Frattini's argument
- Normal Hall satisfies transfer condition
- Equivalence of definitions of finite nilpotent group
- Center of normal implies normal
Proof
We prove this claim by induction on the order of . Rather, we reduce the claim to (1) by induction on the order of
.
Given: A group , a normal Hall subgroup
of
.
To prove: has a permutable complement in
.
Proof: Note that for any prime dividing the order of
, a
-Sylow subgroup of
is also
-Sylow in
. Note also that by fact (2),
-Sylow subgroups exist for all primes
.
- Case where there is a prime
dividing the order of
, such that a
-Sylow subgroup
of
is not normal in
: Let
.
by fact (3) (Frattini's argument). Further, by fact (4),
is a normal Hall subgroup of
, so by induction, there exists a subgroup
of
that is a permutable complement in
to
. We claim that
is a permutable complement to
in
:
-
is trivial: Since
and
is trivial, we obtain that
is trivial.
-
: We have
. Also,
. Thus,
. But as observed earlier, by fact (3),
. this forces
.
-
- Case where
is non-abelian and, for every prime
dividing the order of
, a
-Sylow subgroup of
is normal in
: In this case, each of the
-Sylow subgroups of
is normal in
, hence
is a direct product of its Sylow subgroups. Thus,
is nilpotent. In particular, the center
is nontrivial. By fact (6),
is normal in
. Consider the group
.
is a normal Hall subgroup of
, so by induction, there exists a permutable complement
to
in
. Suppose
is the inverse image of
under the quotient map
. Note that
is a proper subgroup of
, since
is a proper subgroup of
by the assumption that
is not abelian.
and
have relatively prime orders, so
is a normal Hall subgroup of
. Thus, there exists a permutable complement
to
in
.
- Case that
is abelian: This is covered by fact (1).