Cyclic normal implies hereditarily normal
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., cyclic normal subgroup) must also satisfy the second subgroup property (i.e., hereditarily normal subgroup)
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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The subgroup property of being a cyclic normal subgroup is stronger than the subgroup property of being a hereditarily normal subgroup. Equivalently, the property of being cyclic normal is a left-hereditary subgroup property.
- Any subgroup of a cyclic group is a characteristic subgroup thereof
- A characteristic subgroup of a normal subgroup is normal
Given: A group , a cyclic normal subgroup , and a subgroup of
To prove: is normal in
Proof: By fact (1), and the given fact that is cyclic, is characteristic in . By fact (2), and the given datum that is normal in , we conclude that is normal in .