Centralizer of derived subgroup has class at most two
From Groupprops
Statement
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 (?).
Related facts
- 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
Proof
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.