Intermediately normal-to-characteristic implies intermediately subnormal-to-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., intermediately normal-to-characteristic subgroup) must also satisfy the second subgroup property (i.e., intermediately subnormal-to-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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Intermediately subnormal-to-normal subgroup
Further information: Intermediately subnormal-to-normal subgroup
A subgroup of a group is termed intermediately subnormal-to-normal if whenever is a subgroup containing such that is 2-subnormal in , then is normal in .
Intermediately normal-to-characteristic subgroup
Further information: Intermediately normal-to-characteristic subgroup
Given: A subgroup of a group , such that is intermediately normal-to-characteristic in .
To prove: For any subgroup of containing , such that is 2-subnormal in , is normal in .
Proof: Since is 2-subnormal in , there exists a subgroup such that .
Now, is a subgroup of , and is normal in . Since is intermediately normal-to-characteristic in , is characteristic in .
Thus, is characteristic in , that in turn is normal in . By fact (1), is normal in , completing the proof.