Normalizer of pronormal implies abnormal
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., pronormal subgroup) must also satisfy the second subgroup property (i.e., subgroup with abnormal normalizer)
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Statement with symbols
Suppose is a pronormal subgroup of . Then, the normalizer is an abnormal subgroup of .
Further information: Pronormal subgroup
A subgroup of a group is termed pronormal if for any , there exists such that . Here, .
Further information: Abnormal subgroup
A subgroup of a group is termed abnormal if for any , .
- Sylow-normalizer implies abnormal: The normalizer of a Sylow subgroup is an abnormal subgroup. In particular, it is self-normalizing, and any subgroup containing it is also self-normalizing.
Given: Group , pronormal subgroup , .
To prove: For any , .
Proof: By the definition of pronormality, there exists such that . Thus, , so , hence . We know that , so .