Centralizer of derived subgroup has class at most two
Suppose is a group and is the Centralizer of commutator subgroup (?) of . Then, is a Group of nilpotency class two (?). In other words, is either an abelian group or a non-abelian group of class two. In particular, is a Class two normal subgroup (?) and a Class two characteristic subgroup (?).
- Commutator subgroup centralizes cyclic normal subgroup
- Commutator subgroup centralizes aut-abelian normal subgroup
- Centralizer of commutator subgroup is hereditarily 2-subnormal
- Abelian Frattini subgroup implies centralizer is critical
Given: A group . .
To prove: is a group of class (at most) two.
Proof: Since centralizes , and , centralizes . Thus, is contained in the center of . Thus, has class at most two.