Derived subgroup centralizes cyclic normal subgroup

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Suppose N is a cyclic normal subgroup of a group G. Then, the derived subgroup [G,G] is contained in the centralizer C_G(N).

Equivalently, since centralizing is a symmetric relation, we can say that N is contained in the centralizer of derived subgroup C_G([G,G]).

Related facts

Related facts about cyclic normal subgroups

Related facts about descent of action

Related facts about containment in the centralizer of commutator subgroup

Other related facts

Facts used

  1. Cyclic implies aut-abelian
  2. Derived subgroup centralizes aut-abelian normal subgroup


The proof follows from facts (1) and (2).