Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith
From Groupprops
Statement
Suppose is a Primitive group (?),
is a Core-free maximal subgroup (?) of
(or, the stabilizer of a point for a faithful primitive group action of
), and
is an Abelian normal subgroup (?) of
that is a Contranormal subgroup (?) of
:
is generated by the conjugates of
in it. Then, the Commutator subgroup (?)
is the monolith of
, i.e., every nontrivial normal subgroup of
contains the commutator subgroup of
.
Facts used
Proof
Given: A group , a core-free maximal subgroup
of
. An abelian normal subgroup
of
that is contranormal in
. A nontrivial normal subgroup
of
.
To prove: is abelian.
Proof:
-
: Since
is core-free and
is nontrivial normal,
does not contain
. Since
is maximal,
.
-
is normal in
: Clearly,
. Also,
is normal in
and
is normal in <mah>G</math>, so
. Thus,
, so
by step (1).
-
: By assumption,
is contranormal. Thus, the only normal subgroup of
containing
is
. So, by step (2),
is normal in
.
-
is abelian: This follows from facts (1) and (2).