Derived subgroup centralizes cyclic normal subgroup: Difference between revisions

Latest revision as of 17:31, 31 December 2011

Statement

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

Quotient group acts on abelian normal subgroup

Related facts about containment in the centralizer of commutator subgroup

Other related facts

Odd-order cyclic group is characteristic in holomorph

Facts used

Proof

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

@@ Line 1: / Line 1: @@
 ==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 of a subgroup|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;1| ]][[centralizer of derived subgroup]] <math>C_G([G,G])</math>.
 ==Related facts==
@@ Line 14: / Line 16: @@
 * [[Quotient group acts on abelian normal subgroup]]
-==Facts used==
+===Related facts about containment in the centralizer of commutator subgroup===
-# [[uses::Cyclic implies aut-abelian]]
+* [[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]]
-==Proof==
+===Other related facts===
-'''Given''': A group <math>G</math>. A cyclic normal subgroup <math>N</math>.
+* [[Odd-order cyclic group is characteristic in holomorph]]
-'''To prove''': <math>[G,G] \le C_G(N)</math>.
+==Facts used==
-'''Proof''': Consider the homomorphism:
+# [[uses::Cyclic implies aut-abelian]]
+# [[uses::Derived subgroup centralizes aut-abelian normal subgroup]]
-<math>\varphi: G \to \operatorname{Aut}(N)</math>
+==Proof==
-given by:
-<math>\varphi(g) = n \mapsto gng^{-1}</math>.
-Note that this map is well-defined because <math>N</math> is normal in <math>G</math>, so <math>\varphi(g)</math> gives an automorphism of <math>N</math> for any <math>g \in G</math>.
-# The kernel of <math>\varphi</math> is <math>C_G(N)</math>: This is by definition of centralizer: <math>C_G(N)</math> is the set of <math>g \in G</math> such that <math>gng^{-1} = n</math> for all <math>n \in N</math>, which is equivalent to being in the kernel of <math>\varphi</math>.
+The proof follows from facts (1) and (2).
-# <math>\operatorname{Aut}(N)</math> is abelian: This is fact (1).
-# The kernel of <math>\varphi</math> contains <math>[G,G]</math>: Since <math>\varphi</math> is a homomorphism to an abelian group, <math>\varphi([g,h]) = [\varphi(g),\varphi(h)]</math> is the identity. Thus, every commutator lies in the kernel of <math>\varphi</math>, so <math>[G,G]</math> is in the kernel of <math>\varphi</math>.
-# <math>[G,G] \le C_G(N)</math>: This follows by combining facts (1) and (3).