Derived subgroup centralizes cyclic normal subgroup
(Redirected from Commutator subgroup centralizes cyclic normal subgroup)
Equivalently, since centralizing is a symmetric relation, we can say that is contained in the centralizer of derived subgroup .
Related facts about cyclic normal subgroups
- Normal of least prime order implies central
- Cyclic normal Sylow subgroup for least prime divisor is central
Related facts about descent of action
Related facts about containment in the centralizer of commutator subgroup
- Derived subgroup centralizes aut-abelian normal subgroup, so any aut-abelian normal subgroup is contained in the centralizer of derived subgroup
- Abelian-quotient abelian normal subgroup is contained in centralizer of derived subgroup
- Abelian subgroup is contained in centralizer of derived subgroup in generalized dihedral group
- Abelian subgroup equals centralizer of derived subgroup in generalized dihedral group unless it is a 2-group of exponent at most four
The proof follows from facts (1) and (2).