Cyclic normal implies potentially verbal in finite: Difference between revisions
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===Statement with symbols=== | ===Statement with symbols=== | ||
Suppose <math>K</math> is a [[finite group]] and <math>H</math> is a [[cyclic normal subgroup]] of <math>K</math>. In other words, <math>H</math> is a [[normal subgroup]] of <math>K</math> that is also a [[cyclic group]]. Then, <math>H</math> is a [[potentially verbal subgroup]] of <math>G</math>: there exists a group <math>G</math> containing <math>K</math> such that <math>H</math> is a [[verbal subgroup]] of <math>G</math>. | Suppose <math>K</math> is a [[finite group]] and <math>H</math> is a [[cyclic normal subgroup]] of <math>K</math>. In other words, <math>H</math> is a [[normal subgroup]] of <math>K</math> that is also a [[cyclic group]]. Then, <math>H</math> is a [[potentially verbal subgroup]] of <math>G</math>: there exists a group <math>G</math> containing <math>K</math> such that <math>H</math> is a [[verbal subgroup]] of <math>G</math>. In fact, we can choose <math>G</math> to itself be a [[finite group]]. | ||
==Related facts== | ==Related facts== | ||
* [[Cyclic normal implies finite-pi-potentially verbal in finite]] | |||
* [[Homocyclic normal implies finite-pi-potentially fully invariant in finite]] | |||
* [[Central implies potentially verbal in finite]] | * [[Central implies potentially verbal in finite]] | ||
* [[Homocyclic normal implies potentially fully invariant in finite]] | * [[Homocyclic normal implies potentially fully invariant in finite]] | ||
* [[Central implies pi-potentially verbal in finite | * [[Central implies finite-pi-potentially verbal in finite]] | ||
* [[Central implies pi-potentially characteristic in finite pi- | * [[Central implies finite-pi-potentially characteristic in finite]] | ||
==Facts used== | |||
# [[uses::Cyclic normal implies finite-pi-potentially verbal in finite]] | |||
==Proof== | |||
The proof follows directly from fact (1), which is a somewhat stronger formulation. | |||
Latest revision as of 14:38, 20 October 2009
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Cyclic normal subgroup (?)) must also satisfy the second subgroup property (i.e., Potentially verbal subgroup (?)). In other words, every cyclic normal subgroup of finite group is a potentially verbal subgroup of finite group.
View all subgroup property implications in finite groupsView all subgroup property non-implications in finite groups
Statement
Statement with symbols
Suppose is a finite group and is a cyclic normal subgroup of . In other words, is a normal subgroup of that is also a cyclic group. Then, is a potentially verbal subgroup of : there exists a group containing such that is a verbal subgroup of . In fact, we can choose to itself be a finite group.
Related facts
- Cyclic normal implies finite-pi-potentially verbal in finite
- Homocyclic normal implies finite-pi-potentially fully invariant in finite
- Central implies potentially verbal in finite
- Homocyclic normal implies potentially fully invariant in finite
- Central implies finite-pi-potentially verbal in finite
- Central implies finite-pi-potentially characteristic in finite
Facts used
Proof
The proof follows directly from fact (1), which is a somewhat stronger formulation.