Derived subgroup centralizes cyclic normal subgroup: Difference between revisions
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==Statement== | ==Statement== | ||
Suppose <math>N</math> is a [[fact about::cyclic normal subgroup]] of a group <math>G</math>. Then, the [[ | 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 | 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== | ==Related facts== | ||
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===Related facts about containment in the centralizer of commutator subgroup=== | ===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 | * [[Abelian-quotient abelian normal subgroup is contained in centralizer of derived subgroup]] | ||
* [[Abelian subgroup is contained in centralizer of | * [[Abelian subgroup is contained in centralizer of derived subgroup in generalized dihedral group]] | ||
* [[Abelian subgroup equals centralizer of | * [[Abelian subgroup equals centralizer of derived subgroup in generalized dihedral group unless it is a 2-group of exponent at most four]] | ||
===Other related facts=== | ===Other related facts=== | ||
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# [[uses::Cyclic implies aut-abelian]] | # [[uses::Cyclic implies aut-abelian]] | ||
# [[uses:: | # [[uses::Derived subgroup centralizes aut-abelian normal subgroup]] | ||
==Proof== | ==Proof== | ||
The proof follows from facts (1) and (2). | The proof follows from facts (1) and (2). | ||
Latest revision as of 17:31, 31 December 2011
Statement
Suppose is a cyclic normal subgroup of a group . Then, the derived subgroup is contained in the centralizer .
Equivalently, since centralizing is a symmetric relation, we can say that is contained in the centralizer of derived subgroup .
Related facts
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
Facts used
Proof
The proof follows from facts (1) and (2).