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 [[commutator subgroup]] <math>[G,G]</math> is contained in the [[fact about:: | 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>. | ||
Equivalently, since centralizing is a symmetric relation, we can say that <math>N</math> is contained in the [[fact about::centralizer of commutator 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 commutator subgroup]] <math>C_G([G,G])</math>. | ||
Revision as of 13:17, 2 September 2009
Statement
Suppose is a Cyclic normal subgroup (?) of a group . Then, the commutator subgroup is contained in the Centralizer (?) .
Equivalently, since centralizing is a symmetric relation, we can say that is contained in the Centralizer of commutator 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
Facts used
Proof
The proof follows from facts (1) and (2).