Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith
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 .
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.
- : 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).