Every subgroup is contracharacteristic in its normal closure
This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Contracharacteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., 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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Further information: Contracharacteristic subgroup
A subgroup of a group is termed contracharacteristic if it is not contained in any proper characteristic subgroup.
- Characteristic of normal implies normal
- Equivalence of definitions of subgroup of Abelian normal subgroup
Given: A subgroup , is the normal closure of in .
To prove: If is a characteristic subgroup of containing , then .
- By fact (1), we see that since is characteristic in and is normal in , we obtain that is normal in .
- Thus, is a normal subgroup of containing . By definition of normal closure, we get that . Since , we get .