Derived subgroup centralizes cyclic normal subgroup: Difference between revisions

Revision as of 17:25, 31 December 2011

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 $C_{G} ([G, G])$ .

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

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 ==Statement==
-Suppose <math>N</math> is a [[fact about::cyclic normal subgroup]] of a group <math>G</math>. Then, the [[commutator subgroup]] <math>[G,G]</math> is contained in the [[fact about::centralizer]] <math>C_G(N)</math>.
+Suppose <math>N</math> is a [[fact about::cyclic normal subgroup;1| ]][[cyclic normal subgroup]] of a group <math>G</math>. Then, the [[derived subgroup]] <math>[G,G]</math> is contained in the [[fact about::centralizer;2| ]][[centralizer]] <math>C_G(N)</math>.
-Equivalently, since centralizing is a symmetric relation, we can say that <math>N</math> is contained in the [[fact about::centralizer of derived subgroup]] <math>C_G([G,G])</math>.
+Equivalently, since centralizing is a symmetric relation, we can say that <math>N</math> is contained in the [[fact about::centralizer of derived subgroup;1| ]][[centralizer of derived subgroup]] <math>C_G([G,G])</math>.
 ==Related facts==